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Proceedings of the Indian Association for the 
Cultivation of Science. 



Vol. VI. 



CALCUTTA : 

Printed at the Baptist Mission Press and Published by the Indian 

Association for the Cultivation of Science, 

210, Bow Bazar Street, Calcutta. 



1920-1921. 



Contents : 

Parts I & II. 

PAGE 

1. On a New Geometrical Theory of the Diffraction 

Figures observed in the Heliometer. By Sisir 
Kumar Mitra, M.Sc. . . . . . . T 

2. Experiments with Mechanically- Played Violins. By 

Prof. r. V. Raman, Af.A. ' 19 

j. Mechanical Illustrations of the Theory of Large Oscilla- 
tions and Combinational Tones. By Bhabonath 
Banerji, M.Sc. . . . . 37 

.j. Some Phenomena of Laminar Diffraction observed 

with Mica. By Phanindra Nath Ghosh, M.A. .. 51 

5. On the Forced Oscillations of Stretched Strings under 

Damping proportional to the Square of the Velocity. 

By Rajendra Nath Ghosh, M.Sc. . . . . 67 

6. The Magneto-Crystalline Properties of the Indian 

Braunites. By K. Seshagiri Rao, B.A., Hons. . . 87 

7. The Free and Forced Convection from Heated Cylin- 

ders in Air. By Bidhuhhusan Ray, M.Sc. . . 95 

8. Experiments on Impact. By A. Venkatasubbaraman^ 

B.A . . . . . . . . 109 

Parts III & IV. 

9. The Theory of the Flute. By Dr. G, T. Walker, 

C.S.I., D.Sc., F.R.S. .- .. ., 113 

10. On Wave-Propagation* in Optically Heterogeneous 

Media, and the Phenomena observed in Christian- 
sen's Experiment. By Nihal Karan Sethi, M .Sc. 121 

11. On the Production of Musical Sounds from Heated 

Metals. By B. N. Chuckcrbutty, M.Sc. . . 143 

12. Some New Illustrations of Optical Theory by Ripple 

Motion. By Rajendra Nath Ghosh, M.Sc . . 155 

13. Theory of Impact on Elastic Plates. By K. Sesha^iri 

Rao, M.A. . . . . . . . . 165 

14. On Ripples of Finite Amplitude. By J. C. Kames- 

vara Rav, M.Sc. . . . , . . 175 



IV CONTENTS. 

PAGE 

15. On the Effects of a Magnetic Field on the General 

Spectrum. By H. P. Waran, M.A. .. .. 195 

16, On an Automatic Mercury Pump. By H. P. Waran, 

M.A . . . . . . . . . 199 

Index . . . . . . . . 205 



L On a New Geometrical Theory of the 
Diffraction-Figures observed in the 
Heliometer. 



By Slsir Kumar Mitra, M.Sc., Lecturer on Physical Optics in 
the University of Calcutta. 



(Plate I). 



CONTENTS. 

SECTION I. Introduction. 

SECTION II. Principle of the Geometrical Theory. 

SECTION III. Intensity in the Diffraction- Pattern in the two Principal 

Directions. 

SECTION IV. General Configuration of the Pattern. 
SECTION V. The Determination of the Intensity of Illumination at any 

Point in the Pattern. 
SECTION VI. Synopsis. 



SECTION I. INTRODUCTION. 

A mathematical theory of the form of the diffraction figures 
of the Fraunhofer class due to a semi-circular aperture was first 
developed by Bruns * who gave the formulae for the intensity of 
the light at any point in the neighbourhood of the focus of a 
heliometer objective. These formulae, however, demanded a very 
large amount of arithmetical calculation before they could be of 
any practical use in the determination of the form of the diffrac- 
tion pattern, and the further work involved in this appears to 



* Bruns, * Uber die Beugungsfiguren des Heliometer-Objectives," * Astr. 
Nachr.,' Bd. CIV, No. 2743. 
B 



2 S. K. MlTRA. 

have remained unattempted till 1909 when Everitt * succeeded in 
carrying out the numerical computations and finding the form of 
the contour-lines of equal illumination in the neighbourhood of the 
focus in the diffraction pattern. The process adopted by him 
was extremely elaborate, and involved the use of a Brunsviga 
calculator for part of the numerical work, of a Coradi co-ordi- 
natograph, and of an integrator and planimcter for semi-graphical 
and mechanical integration. While Everitt 's paper is of undoubted 
value in view of the painstaking thoroughness and accuracy 
of his work, the method adopted by him is unsatisf actorv 
from a physical point of view, as it is essentially numerical in 
spirit, and leaves the reader without a clear conception of how 
precisely the peculiar configuration of the pattern arises from the 
semi-circular form of the aperture. Everitt's work was also of 
somewhat limited scope, as the calculation of intensities was con- 
fined to a relatively small region surrounding the position of the 
geometrical focus. 

It will be shown in the present paper how the theoretical 
determination of the form of the diffraction figures in the helio- 
meter can be carried out without appreciable loss of accuracy by 
a method which is geometrical in character and is far less laborious 
than that adopted by Everitt. Not merely is the method simpler, 
but it has also the advantage of enabling the form of the pattern 
to be readily determined for a region of any desired area surround- 
ing the focus. The geometrical form of the pattern deduced by 
this method is in close agreement with experiment, and shows 
certain features at large angular deviations from the focus which 
were unsuspected by Everitt and which indeed make necessary a 
modification of the description of the diffraction-figures published 

by him. 

To illustrate the novel features observed by the author, photo- 
graphs of the diffraction pattern with relatively long exposures 
have been secured, one of which enlarged to a suitable size is re- 
produced in Fig. i (Plate). For purpose of comparison, a photo- 
graph taken with a comparatively small exposure is reproduced in 



* Everitt, ' Diffraction Figures due to the Heliometer," ' Proc. of the Royal 
Society of Lond.' Series A, Vol. 83; 1910. 



K. MITRA. 



PLATE !. 




Fig. 



Fig. 2 



Diffraction Figures observed in tfU.Heliometer. 



A New Theory of the Heliomeier Diffraction-Figures. 3 

Fig. 2, the picture in this case being very similar to those published 
in Everitt's paper.* It will be noticed that Fig. i (Plate) shows 
the prolongation of the transverse streamers as concave outwards, 
a feature which fails to appear in the relatively limited region 
covered by the picture in Fig. 2 (Plate). The work has also 
brought to light another important feature not mentioned by 
Everitt, that the fluctuations of intensity along the long horizontal 
ray that appears crossing the pattern decrease both relatively and 
absolutely at large angular deviations. 

vSECTiON II. PRINCIPLE OF THE GEOMETRICAL THEORY. 

The ordinary treatment of diffraction phenomena, both of the 
Fresnel and of the Fraunhofer class, proceeds by expressing the 
effect at any point in the field in terms of a surface integral taken 
over the area of the aperture, in other words, as the resultant of 
a collection of secondary sources of light situated over the whole 
area of the boundary. A considerable simplification is, however, 
effected by transforming the surface integral into a line integral, 
in other words, by regarding the effect at any point in the field 
as due to a linear source of light situated along the boundary of 
the aperture, and the linear source in turn can be replaced by a 
finite number of point sources of light, generally two, sometimes 
three or more, having appropriate phases and situated at certain 
points on the boundary. The position and intensity of these 
point-sources on the boundary is generally not fixed, but varies 
with the direction of the diffracted light. In other words, cor- 
responding to each point in the focal plane at which the diffraction 
pattern is formed, we have certain points on the boundary which 
principally contribute to the luminous effect at the point of ob- 
servation, and the whole of the diffraction pattern may be simply 
regarded as an interference pattern clue to a finite number of 
light-sources of variable position situated on the boundary of the 
aperture. 

The foregoing method of regarding diffraction is of con- 



* And also in i recent paper by Gordon, Proc. of the Phys. Soc. of Lond., 
>ber 1012. 



October 1912. 



4 S. K. MITRA. 

siderable utility in forming a mental picture of the way in which the 
phenomena due to any aperture of arbitary form arise, and ex- 
plaining the geometrical relationship between the form of the aper- 
ture and the form of the diffraction pattern produced by it. This 
has been emphasised in a recent paper by me on large-angle 
diffraction by curvilinear boundaries.* So far from being merely 
a convenient mathematical fiction, the existence of sources of light 
situated at specific points on a curvilinear diffracting boundary 
may be directly verified by observation or photography, as has 
been shown in a recent paper. f For this purpose, the aperture 
may be viewed by the aid of the diffracted light only, admitted 
into an observing telescope through a small hole in a screen other- 
wise completely cutting off the light reaching the focal plane [as 
in the well-known method of the Foucalt test). When the lumi- 
nous emission from the boundary of a semi-circular aperture is 
observed in the foregoing manner, the following phenomena are 
noticed : in general, three points on the boundary are seen to be 
luminous: two of equal but small intensity are situated on the 
two corners, and a third and more intense one is situated on the 
arc of the semi-circle at such point that the corresponding radius 
of the semi-circle is parallel to the line joining the centre of the 
focal plane with the orifice through which the diffracted light enters 
the observing telescope. There are, however, two exceptions to 
this general rule : when the orifice is situated at any point on the 
long horizontal ray in the diffraction-pattern running perpendi- 
cular to the diameter of the semi-circle, the whole of the diameter 
as well as the mid-point of the arc appear luminous. On the 
other hand, if the orifice in the focal plane is situated on a line 
drawn parallel to the diameter of the semi-circle, we have only 
two light sources visible which are situated respectively at the two 
extremities of the diameter and appear brilliantly luminous. The 
transition from the phenomena as observed in the two special 
portions of the orifice in the focal plane to that seen in the general 
case when it is placed in any arbitary position takes place in a 
fairly sudden manner. 



* S. K. Mitra, Phil. Mag., Sept. 1919. 

t Dr. S. K. Banerjee, Phil. Mag., Jan. 1919. 



A New Theory of the Heliometer Diffraction-Figures. 5 

SECTION III. INTENSITY IN THE DIFFRACTION-PATTERN 
IN THE TWO PRINCIPAL DIRECTIONS. 

The illumination at any point in the focal plane lying in 
directions parallel and perpendicular to the diameter of the semi- 
circular aperture may be very readily calculated as follows : 

Case I : Parallel to the diameter. 

In this case we may begin by dividing the semi-circle into 
narrow strips parallel to the diameter and adding up the effects of 
these strips. 

To obtain the effect of one of these strips we observe that 
the expression for disturbance due to an elementary portion dx dy 
of the strip (of width dy) may be written as 



A ' sin I wt x } dx dy, 

\ A / / 



(the X axis being taken parallel to the strips), where r is the 
distance of the point of observation from the centre of the pattern 
in the focal plane, / the focal length of the lens, and A' is a const. 

such that if I be the intensity at the centre of the pattern, \/I=*A ' X 
area of the aperture. The above when integrated between the 
limits x y and x i) the co-ordinates of the two extremities of the strip 
gives us 




where 

A I 

* 

the intensity at the centre of the pattern being taken to be unity. 
R is the radius of the semicircular arc. 

This shows that the effect of each strip may be replaced by 
that of two light-sources of equal strength and opposite phase 
situated at its extremities. (It will be noticed that the strength 
of the sources is independent of the length of the strip.) The 
effect of the whole aperture accordingly reduces to a linear dis- 
tribution of light sources along the semicircular arc. If the aper- 



6 S. K. MITRA. 

ture had been a complete circle instead of being a semicircle, there 
would have been a precisely similar and symmetrical distribution 
along the other half of the circle. It follows from this that the 
resultant amplitude at any point due to the distribution of sources 
on the boundary of the semicircular aperture is half that for the 
case of the circular aperture, the intensity being one-fourth. The 
positions of the maxima and minima of illumination along this direc- 
tion are accordingly identical with those obtained in the well-known 
investigation for the case of a circular aperture. 

Case II : Perpendicular to the diameter. 

In this case, the semicircular aperture may be imagined 
divided up into strips perpendicular to the diameter. Proceeding 
in the same manner as before, the effect of the whole aperture 
reduces to a linear distribution of sources along the straight and 
the curved parts of the boundary, which it is convenient to con- 
sider separately. 

For the straight portion of the boundary we have a distribu- 
tion of sources of amplitude 

zdv . 



The X axis being taken parallel to the strips (i.e. perpendicular 
to the diameter), X L is constant, and the phases of the contribu- 
tions of the different parts of the boundary are all the same. The 
effect due to the whole diameter is thus obviously 



or taking the Y axis along the diameter (i.e. #,=o), it is equal to 

4 / "\ 

sin I wt-- I 

718 V 27 

For the curved portion of the boundary we have a similar 
distribution of sources of amplitude 

'2,dy . / 2?r r TT\ 

sm ( w i- TJ - x ,+- 2 ). 

In this case, however, x being different for different parts of 
the boundary, the phases of the contributions from different parts 



A New Theory of the Heliometer Diffraction-Figures. j 

would vary rapidly and would be stationary only at the mid-point 
of the arc. The resultant effect of the sources situated along the 
semicircular arc may be deduced at once from the consideration 
that if a precisely similar distribution of sources on a semicircular 
ate be placed symmetrically on the other side of the diameter, the 
joint effect of the two together should give us the well-known 
expression for the case of a complete circular aperture which is 

4 -- sin wt * 

o 

Using semi-convergent expansions we have 




approximately, and the expression for the disturbance due to the 
complete circular boundary is accordingly 



i/I 

o v Trrt 



sin* 

4 



This disturbance may be- resolved into two parts, 

2/2 



and <iz* 

each due to one-half of the circular boundary. The effect due to 
the distribution of sources situated along the semicircular arc in 
this case is thus equal to 



* This is usually written _> --- sin wt, taking the intensity at the centre of 

5 

the pattern due to the circular aperture to be unity. Since we have taken the 
intensity at the centre of the pattern due to the semicircular aperture to be 



unity the expression is 4 *- sin wt. 
$ 



8 



S. K. MITRA. 



The complete expression for the disturbance due to the semi- 
circular aperture is the sum of that due to the curved and straight 
parts of the boundary, that is equal to 



It is obvious that of these two terms, the latter varying 
as 3~ ! ultimately pieclominatcs for increasing values of 8. The 
intensity in the diffraction pattern in the direction perpendicular 
to the diameter thus ultimately varies as S~ 2 , while in a direction 
parallel to the diameter, it decreases much more rapidly in the 
ratio of S~ 8 . Thus, we have in the diffraction pattern a long 
bright ray running perpendicular to the diameter of the semi- 
circle. Further, in the direction of the bright ray, not merely do 
the absolute intensities diminish, but the ratio of the intensities 
at maxima and minima also tends to approach unity, in other 



Centre 
ist Min. 
ist Max. 
2nd Mm. 
2nd Max 
3rd Min. 
3rd Max. 
4th Mm. 
4th Max. 
5th Min. 
5th Max. 
6th Min. 
6th Max. 
7th Mm. 
7th Max. 
8th Min. 
8th Max. 
9th Min. 
9th Max. 
loth Min. 
loth Max. 
nth Min. 
nth Max. 



Author's Values. 


Everitt's Values. 


5 


H fi 


H 


- 




i 


O'O 


I (XX) O'O 


I'OOO 


7 < >o 


o -00897 , 7-1 


0-009 


g-65 


0-0324 97 


0-033 


13-40 


0-00392 


iV4 0-00397 


16*09 


0*01050 


16*0 i 001057 


T9-69 


o 00216 


iQ*4<;* 


o 00257* 


22-47 


0-00506 


22*45 


0-00503 


2;-<) 


of 10137 


2-VQ5 


0-00137 


28-80 


O*(X>2Q4 


28-85 


0-00296 


32*26 


0*00094 


32-25 


0*00095 


WiS 


o 00191 


35-15 


0*00193 


3'54 


0-00069 


38-qo 


0*00068 


41-42 


0-001342 






44*83 


0-000535 






4773 0-000980 






51-11 


0-000422 






; 54'3 


0*000763 




57'39 


o ooc >34O 




60-32 


o -0005 99 




; <>v<>7 


0-000284 




66-61 


0-0x30469 I 


69-95 


0-000240 | 




72*92 


0-000402 


i 



* Everitt's results seem to be in error. 



A New Theory of the Heliometer Diffraction-Figures. 9 

words, the fluctuations of intensity along the bright ray tend to 
diminish both relatively and absolutely. The formula given above 
enables the intensity to be readily calculated for any value of 8 
with great ease. In the following table, several values and posi- 
tions of the maxima and minima along the bright ray are shown, 
and alongside for comparison Everitt's values for the first few 
maxima and minima are also given. It will be seen that the same 
degree of accuracy as that obtained by Everitt is obtainable by 
this method with far less labour. 

vSEcxiON IV. GENERAI, CONFIGURATION OF THE PATTERN. 

We now proceed to prove that (excluding the two principal 
directions) the whole effect of the aperture might be regarded as 
equivalent to three sources of appropriate phase and intensity 
two of which are situated at the two corners and the other on the 



or' 





FIG. 3. 

semicircular arc. The general configuration of the pattern will 
also be deduced on this basis. 

Divide the semi-circular aperture into strips parallel to the 
line joining the point of observation to the centre of the pattern 
in the focal plane (as for instance parallel to OB in Fig. 3). As 



10 S. K. MlTRA. 

before, the effect of all these strips can be reduced to a linear dis- 
tribution of sources situated on the boundary of the aperture. 
The phases of the contributions from different parts of the straight 
portion of the boundary change regularly from C to D. We 
can accordingly replace the effect by a source situated at the 
centre 0, or by two sources of opposite phase and equal amplitude 
situated at C and D. The phase of the resultant source at 
being - w / 2 (the elementary contributions being retarded by that 
amount on account of strip division), the phases of the component 
sources at C and D would be -8 sin and & sin 0-v (the phases 
of the contributions from C and D are 8 sin and 8 sin d respec- 
tively, if the phase- of contribution from is taken to be zero, 
that from 13 being - S. 0=*/z - +.COU). For the curved 
portion of the boundary, produce the semi-circle DEC to meet 
C'D' drawn perpendicular to OB. We observe that the effect of 
the arc CKD is obviously equivalent to the effect of 

(the semicircular arc D'BC') (the arc C"C) + (the arc DD'). 

The semicircular arc C'B'D' y we have already seen, can be re- 
placed by a source of proper phase ( - ^ +-*) and amplitude 
situated at B. The effect of the arcs CC' and DD' are replaceable 
by two sources at C and D respectively of equal amplitude (since 
the arcs are of equal length), whose phases are the same as the 
phases of the components at these points due to the straight 
portion of the boundary CD.* We thus have (or the whole aper- 

* This is evident from the following considerations : The amplitude of an 
elementary portion ds of the boundary being proportional to the width of the 
corresponding strip, is proportional to ds cos ^ (\J/ = angle between the normal to ds 
and the length of the strip). The amplitude of an elementary contribution con- 
sequently changes from a maximum at B to zero value at C, while the phase 
advances from 8 to 5 sin tf. If such a system of vibrations of continually 
decreasing amplitude and advancing phase is compounded, the phase of the 

resultant is approximately the same as that of the first vibration plus *. (This 

result is always true except when the phase of the first vibration passes through 
a maximum or minimum value. The above arguments are easily understood if the 
vibrations are compounded graphically. Cf. Preston, Theory of lyight (third edi- 
tion), page 250, fig. 1 20. The resultant effect due to the two arcs are therefore two 
sources at C and D of phases 3 sin t> + w/a and 5 sin ir/.j (unless C or D He close to 
B , where the phase is a minimum). Remembering that the phases of the elementary 
portions of the arcs CC' and DD'are advanced and retarded respectively by w/3(on 



A New Theory of the Heliometer Diffraction-Figures. n 

ture a source at B (due to the semicircular arc D'BC') y and a pair 
of sources at C and D (due both to the straight portion CD and 
the arcs CC' and DD') } their phases being 

- 8 + 77 at B 

-8 sin at C 

8 sin 0-7T at />. 

(The source B is of course of variable position : its point of 
situation on DEC depending on the point of observation in the 
focal plane as explained before). 

In order to find the nature of the diffraction pattern let us 
take any two of the sources and find out the positions of constant 
phase difference in the focal plane. Thus, for the sources C and B 
there would be maximum of illumination when 

& sin + S |^ = 2w, 4?r, etc., i.e. 8~ 8 sin = const. 

This would give us in the focal plane a series of parabolas 
branching upwards, of common axis lying parallel to the diameter 
of the semicircle. Similarly the sources D and B would give us 
another set of parabolas, their axis the same as the former set, but 
the parabolas themselves branching downward. 

8 + 8 sin $ = const. 

Also for the sources C and D y the condition of the maximum 
illumination is 

8 sin # + & sin #-7r = 27r, 477, etc., i.e. 8 sin = const. 

This gives us a number of straight lines in the focal plane 
running perpendicular to the diameter of the semi- circle, i.e. 
parallel to the horizontal bright ray. These features, viz. two 
sets of parabolas branching upwards and downwards, and a set of 
straight lines (a few of them) running perpendicular to the axis of 
the parabolas are shown in the drawings, Figs. 4 and 5. It will be 
seen that Fig. 4 closely reproduces the features appearing in the 
long-exposure photograph of the author in the Plate. (Fig. i). 

There is an alternative way of regarding the " interference " 



account of strip division), and that the effect of CC' is to be subtracted, the 
phases of the resultants at C and D finally become 8 sin o and 8 sin *. 



12 S. K. MlTKA. 

pattern in the focal plane which is both useful and instructive. 
The phase of the two sources at C and D being 

6 sin and 8 sin n, 

these when combined together give a source of fluctuating am- 
plitude * at proportional to sin (8 sin 6), and phase - - . 

The " interference " pattern in the focal plane can be regarded 
as due to the superposition of a source of fluctuating amplitude 
at 0, and a source of variable position at B on the semicircular arc 
(the line OB being parallel to the line joining the point of observa- 
tion in the focal plane to the centre of the pattern). Since the 




phases of these sources are independent of (i.e. direction of 
observation in the focal plane), the lines of equal phase will be 
circles round the centre of the pattern. The real maxima will 
however be at those positions in the focal plane where both and 
B agree in phase, and at the same time the amplitude of the source 
is a maximum (the amplitude of the other source being in- 
dependent of 0). 

* The part of the amplitude of the source at O due to the straight portion is 

as will be seen in the following section really proportional to : , so that the 

8 sin o 

positions of the maxima are slightly different from that given by 6 sin 0=(2n + 1)~* 

when is small. This has been taken into account in drawing fig. 5. For 
small values of the other part of the amplitude (due to the arcs CC' and DD') is 
very small, so that the positions of maxima coincides with the maximum values 
sin (5 sin 6) 
5 sin 



A New Theory of the Heliometer Diffraction-Figures. 13 

The first condition is given by 



and the second by 



8 sin 0=- , 5 - , o - , etc. 
2 2 2 



The first gives us a series of circles, and the second a series of 
parallel straight lines. The loci of maximum illumination lie 
on the intersections of these two sets of curves and are obviously 
parabolas. Fig. 5 shows the forms of the curves as obtained from 
the above considerations. The features appearing in the drawing 




FIG. 5. 

are also clearly noticeable in the long-exposure photograph in the 
Plate. 

SECTION V. THE DETERMINATION OF THE INTENSITY OF 
ILLUMINATION AT ANY POINT OF THE PATTERN. 

The effect at any point in the focal plane being equal to the 
sum of two sources, one at and another at a corresponding point 
B on the arc, we proceed to find the amplitude of these sources. 

The amplitude of the source at B due to the semicircular arc 
D'BC', we have already found to be 



2 V- 

8V 7r8 ' 



14 S. K. MITRA. 

that of the source at O being composed of two parts, one due to 
the straight portion CD and the other due to the arcs CC' and >D X , 
we can find their resultants separately. 

The contribution from an elementary portion ds of CD being 



zds . . / 2-rr s . sin 9 



TT\ 

-a) 



its total effect is found by integrating the above expression 
between limits 7? and R (vide Fig. 3. OC OD R and s== 
distance of ds from 0) to be 



4 - sin (8 siii 0) . 
JL cos 9 \ sin 

*8 S sm (9 



/ W\ 

in I wt - J . 



The resultant effect due to the arc CC' (or DD') can be found 
by dividing it into half-period (l zones" and adding up the effects 
of all of these. This can be done in the following way : 

Beginning from B (fig. 3) divide the arc BC/ into small parts 
such that the phases of the resultant contributions from successive 
parts (with reference to the point of observation in the focal plane) 
differ by *-, that of the first one being the sa*ne as the resultant of 
the whole arc BC' . This is easily done by following a method 
exactly analogous to that employed by Schuster * in finding the 
resultant effect of a plane wave, and the total effect of the arc 
BC', i.e. half the semicircular arc (except the first 'zone') can be 
expressed in the form 

2 /~ - __,_- 

" / Ji_ I (\/7 \/ ]) (vi i v'y + (v'l 5 \/ ii ) + I 

8 v 7r8 

Knowing the effect due to half the semicircle to be 

2 /7 - 

^v ^* > 

the effect of the first f zone ', since it is of opposite sign to that 
of the resultant of the rest of the arc, is easily seen to be 

6 2~x 2 /I 
^7 2 5 x ^ v/ w5 

* Schuster, Theory of Optics (2nd edition, 1909), page 95. 



A New Theory of the H diameter Diffraction-Figures. 15 



the sum of the above series within bracket being '1725. The effect 
of half the semicircular arc can thus be expressed in the form 



2 /!< 

sv *s v 



6725 - -2908 + -2135 --1771 etc.). 



By omitting the first term or the first two terms or the first three 
terms from the above series, the effect of corresponding portion of 
the arc as given in the following table can be found out : 



01- THE ARC. 



Half-semicircle 



83 sin f= 



except ist zone 
8 5 sin 0=3 - 



except first two zones 



except first three ,, 

HIT 
5 5 sm e= 



etc. 



AMPLITUDE. 



i/25* 



002 5 x 



etc. 



If a curve be now plotted between different values of 
8-8 sin & (corresponding to different lengths of the arc) and the 
corresponding effect in amplitude as found from the above table, 
we can obtain from that the effect in amplitude for any value of 
8-8 sin 9, i.e. for any length of the arc CC/ (or DD'). Designating 
the amplitude thus found by 

,./(<*-8 sin 0), 

b 

the expressions for the resultant sources at C or D due to the arcs 
CC' and DD 1 taking their proper phases into account as found in 
4 are at once obtained. 



-/(S-Ssm 0) sin (wf-8sin 0) 



i6 
and 



vS. K. MlTRA. 



^./-^/(S-Ssintf) sin (zetf + 8 sinfl-ir). 

o V rrrt 

These when combined give us a resultant source 

2 /T I \ I 7T 

2 /(8 -d sin 0) - / sin I 8 sin 6 ] sin I wl - - 
8 v 718 V / \ 2 



at0. 

We thus have the complete expression for amplitude for any 
point in the focal plane : 



M -T ( 



In the following table are given the values of intensity H for 
various values of and <$, in the focal plane together with Everitt 's 
values for comparison. The latter were obtained from his contour 
diagram of equal intensity. 

The agreement can be seen to be satisfactory. 



8 = 7-01 








9V 


33 


50 


63 


73 


90 


Author . . 


H i -0092 


0050 


0163 


022 


007 


0033 


oooo 


Everitt . . 


H 0098 


004 


015 


015 


006 


002 


oooo 



Author . . 


a 





I348' 


32 
0007 


4i48' 


55i2' i 70 


90 

OOOI2 


H 


0291 


0068 
006 


0040 
004 


0045 , -0015 


Everitt . . 


H 


0291 


ooi 


004 j *ooi 


oooi i 



A New Theory of the Heliometer Diffraction-Figures. 17 



8=13-32 



Author 



Everitt 




H 








I548' 


286' 


3942' 


50*48' 


6oi8' 


6930' 


80*24' 




0039 


'0021 


ooio 


00038 


0020 


O022 


0012 


00036 


004 


'002 


ooi 


0005 


'002 


002 


ooi 


'0002 



90 



0000 



8=16-47 



Author 
Everitt 



H 



H 






530' 


21*48' 


3'24 


0102 


0059 


.00052 


ooi 


OlOj 


006 


0005 


ooi 



3'24' 


4 848' 


56 


63V 


oo' 


ooi i 


00049 


ooio 


ooi i 


oooo 


ooi 


0005 


ooi 


ooi 


oooo 



SECTION VI. SYNOPSIS. 

1. Photographs of the diffraction pattern due to a heliometer 
objective with relatively long exposures have been secured which 
show certain features hitherto unsuspected by previous observers. 
The pattern roughly consists of two sets of parabolas (with a 
common axis, at right angles to the bright horizontal ray) branch- 
ing in opposite directions. These features do not clearly appear 
in the relatively small region covered by the pictures of previous 
workers. Another important feature not mentioned by Everitt, 
who has made a detailed study of the diffraction figures, is that 
the fluctuations of intensity along the long bright ray that ap- 
pears crossing the pattern decrease both relatively and absolutely 
at large angular deviations. 

2. A new geometrical theory has been developed by means 
of which the light intensity at any point in the focal plane is con- 
sidered as merely due to the interference of two or three light 
sources situated at definite points on the boundary. This is done 
by a simple transformation of the ordinary expression for the 
light vector taken as an integral over the surface of the aperture, 
into a line integral taken round the boundary, which latter is 

D 



1 8 S. K. MITRA : Heliometer Diffraction-Figures. 

finally reduced to two or three sources of proper phase and ampli- 
tude situated on the boundary of the aperture. This method 
greatly facilitates numerical computation of the light intensity for 
small as well as for very large angles of diffraction (the latter 
being almost impossible to obtain by the ordinary treatment), and 
gives results , which are for all practical purposes, as accurate as 
that obtained by elaborate mathematical formulae. It has also 
the additional advantage of enabling the form of the pattern to 
be readily determined for any desired area surrounding the focus, 
and the geometrical form thus deduced is in close agreement with 
that determined experimentally. 

3. The writer has made preliminary observations on the dif- 
fraction figures due to apertures greater and less than a semicircle 
respectively, and has found that in these cases also the form of 
the pattern can be deduced from the above simple geometric con- 
siderations. 

The investigation was carried out in the Palit Laboratory of 
Physics, and the writer wishes to thank Prof. C. V. Raman for his 
helpful interest during the progress of the work. 



IL Experiments with Mechanically-Played 

Violins. 



By Prof. C. V. Raman. 



(Plate II.) 



CONTENTS. 

SECTION I. Introduction. 

SECTION II. Description of Mechanical Player. 

SECTION III. Variation of Bowing Pressure with the Position of the 

Bowed Region. 

SECTION IV. Relation between Bowing Speed and Bowing Pressure. 
SECTION V. Variation of Bowing Pressure with Pitch. 
SECTION VI. Effect of Muting on the Bowing Pressure. 
SECTION VII. Other Applications of the Mechanical Player. 
SECTION VIII. Synopsis. 



SECTION I. INTRODUCTION. 

In the first volume (recently published *) of my monograph on 
the theory of the violin family of instruments, I have discussed on 
mechanical principles, the relation between the forces exerted by 
the bow and the steady vibration maintained by it, and the con- 
ditions under which the bow is capable of eliciting a sustained 
musical tone from the instrument. An experimental test of the 
results indicated by the theory on these points would obviously 
be of interest. Especially is this the case, as the analysis shows 
that the yielding of the bridge and the communication of energy 
from the strings through their supports into the instrument and 
thence into the air, play a very large part in determining the 

* Bulletin No. 15 of the Indian Association for the Cultivation of Science, 
1918, pages 1-158. 



20 C. V. RAMAN. 

magnitude of the forces required to be exerted by the bow. An 
experimental study of the mechanical conditions necessary for 
obtaining a steady musical tone could thus be expected not merely 
to throw light on the modus operandi of the bow but also to 
furnish valuable information regarding the instrument itself, its 
characteristics as a resonator and the emission of energy from it 
in various circumstances. Further, a study of the kind referred 
to could be expected also to furnish illustrations of the physical 
laws underlying the technique of the violinist and to put these 
laws on a precise quantitative basis. The experiments described 
in the present paper were undertaken with the objects referred to 
above, and the description of the results now given in these 
Proceedings is preliminary to a more exhaustive treatment of the 
subject which it is proposed to give in the second volume of my 
monograph under preparation for publication as a Bulletin of the 
Association. 

SECTION II. DESCRIPTION OF MECHANICAL PLAYER. 

As the object of the work was to elucidate the theory of pro- 
duction of musical tone from instruments of the violin family, it 
was decided that the experimental conditions should approximate 
as closely as practicable to those obtaining in ordinary musical 
practice. The general principle accordingly held in view in 
designing the mechanical player was to imitate the technique of 
the violinist as closely as possible. There was also another reason 
for adopting this course. It is well known that the bowing of a 
stringed instrument so as to elicit a good musical tone is an art 
requiring much practice for its perfect accomplishment. The per- 
formance of the same task by purely mechanical appliances under 
such conditions as would permit of accurate measurements of the 
pressure and speed of bowing and the discrimination by ear of the 
effect of varying these factors obviously involves difficulties 
which it was thought would be best surmounted by imitating the 
violinist's handling of the bow as closely as the mechanical con- 
ditions would permit. A mechanical player designed on this general 
idea which has fulfilled the requirements of the work is illustrated 
in Plate II. 

As can be seen from the photograph, a violin and a horse-hair 
bow of the ordinary type were used in the mechanical player. 



D 

n 



o^ 

5' 



p_ 

n> 

T 

cT 



o 
c 

CO 



m 

X 



3 

CD 

D 




I 31V1d 



'A 



Mechanically-Played Violins. 21 

Instead, however, of moving the bow to and fro, it was found a 
much simpler matter from the mechanical point of view to keep 
the bow fixed and to move the violin to and fro with uniform 
speed. This was arranged by holding the violin lightly fixed in a 
wooden cradle, the points of support being the neck and the tail 
piece of the violin as in the ordinary playing of the instrument. 
The cradle was mounted on a brass slide which moved to and fro 
noiselessly on a well-oiled cast-iron track. The slide received the 
necessary movement forward and backward from a pin carried by 
a moving endless chain and working in a vertical slot carried by 
the slide. The chain was kept in motion by the rotation of one of 
the two hubs between which it was stretched, this hub being 
fixed on the same axis as the driving wheel seen in the Plate.* 

The apparatus was driven by a belt running over a conical 
pulley which in its turn was driven by a belt passing over the 
pulley of a shunt-wound electric motor controlled by a rheostat 
which was allowed to run without any load except the apparatus. 
Using the rheostat and a Weston Electrical Tachometer, a very 
constant speed could be maintained during the experiments. 
Different speeds of motion of the slide carrying the violin were 
obtained by putting the driving belt of the apparatus on to 
different parts of the conical pulley, or by adjusting the rheostat. 

The mounting of the bow required special attention in order 
to ensure satisfactory results. As is well known, the violinist in 
playing his instrument handles the bow in such manner that when 
it is applied with a light pressure, only a few hairs at the edge 
touch the string. The bow is held carefully balanced in the 
fingers of the right hand, the necessary increases or decreases in the 
pressure of bowing being brought about by increase or decrease of 
the leverage of the fingers. The suppleness of the wrist of the 
player and the flaccidity of the muscles of the fore-arm secures the 
necessary smoothness of touch. These features are carefully 
imitated in the mechanical player. The violin-bow is held fixed 
at the end of a wooden lath, an adjustment being provided so that 
fewer or more hairs of the bow may be made to touch the string 

* The whole of the apparatus was improvised in the laboratory from such 
materials as were to hand. The slide and cast-iron track were parts of a disused 
optical bench. The chain and hubs were spare parts purchased from a cycle- 
dealer. The ball-bearing of the axle of the lever (referred to below) was also part 
of a cycle. The other fittings were made up in the workshop. 



22 C. V. RAMAN. 

of the violin. The lath itself is balanced after the manner of a 
steelyard, the axis of the lever being mounted on ball-bearings so 
as to secure the necessary solidity combined with freedom of move- 
ment. The weight of the bow is balanced by a load hung freely 
near the end of the shorter arm of the lever. The axis of the lever 
can be raised or lowered to the proper height above the violin 
such that when the hairs of the bow touch the string, they are 
perfectly parallel to the cast-iron track along which the violin 
slides. This is of great importance in order to obtain steady bow- 
ing, as otherwise the bow would swing up and down with the 
movement of the violin along the track, and its inertia would 
result in a variation of the pressure exerted by it. Any residual 
oscillations of the bow due to the elasticity of the lever or imper- 
fection in the adjustment referred to above are checked by the 
damping arrangement shown in the Plate. A wire with a number 
of horizontal disks attached to it at intervals is hung freely from 
the shorter arm of the lever and dips inside a beaker of water or 
light oil. This effectually prevents any rapid fluctuations in the 
pressure of the bow and ensures a smooth movement. The pres- 
sure exerted by the bow on the string can be varied by moving a 
rider along the longer arm of the lever which is graduated. An 
adjustment is provided by which the block carrying the axis of the 
lever can be moved by a screw perpendicular to the track, and the 
distance from the violin-bridge of the point at which the bow 
touches the string may thus be expeditiously altered. 

It will be noticed that with the arrangements described 
above, the pressure exerted by the bow on the string of the violin 
would not be absolutely constant throughout, but would vary 
somewhat as the violin moves along its track from the point 
nearest to the point furthest from the axis of the lever. This is 
not however a serious difficulty as the lever is fairly long and the 
variation of pressure is thus not excessive. Further, the observa- 
tions of the character of the tone are always made for a particular 
position and direction of movement of the violin and no ambiguity 
or error due to the cause referred to above arises. 

The speed of bowing may be readily determined from the 
readings of the Electrical Tachometer or directly by noting on a 
stop-watch the time taken for a number of strokes to and fro of 
the violin on its track. 



Mechanically-Played Violins. 23 

SECTION III. VARIATION OF BOWING PRESSURE WITH THE 
POSITION OF THE BOWED REGION. 

One of the well-known resources of the violinist is to bring the 
bow nearer to or to remove it further away from the bridge of the 
violin, the extreme variation in the position of the bow being from 
about 1th to about aVth of the vibrating length of the string from 
the bridge. In a recent paper on " The Partial Tones of Bowed 
Stringed Instruments" published in the " Philosophical Magazine" 
(November 1919), I have discussed in some detail the changes in 
the amplitudes and phases of the various partials brought about 
by these changes in the position of the bowed region. In all the 
cases of musical interest within these limits, the mode of vibration 
of the string is practically the same as in the principal Helmholt- 
zian type * in regard to the first three partial components, but 
differs from it in respect of the higher components to an extent 
depending on the removal of the bow from the bridge. The 
ratios of the amplitudes and the relative phases of the first three 
partials remain practically the same throughout the range, the 
actual amplitudes for a given speed of the bow varying inversely 
as the distance of the bowed point from the bridge. The ampli- 
tudes of the fourth, fifth and higher partials vary in a similar way 
with the position of the bowed point provided this is not too far 
from the bridge, but deviate from this law more and more as the 
bow is removed further and further from the bridge. The net 
effect of bringing the bow nearer the bridge (its speed remaining 
constant) is greatly to increase the intensity of the tone of the 
instrument, and to make it somewhat more brilliant in character, 
as is of course well known. Simultaneously with these changes, 
the pressure with which the bow is applied has to be increased. 
The mechanical player described above may be used to find ex- 
perimentally the relation between the bowing pressure and the 
position of the bow under these conditions. 

The graphs in Fig. i (thin lines) represent the results obtained 
with the player on the D-string of the violin. A few words of 
explanation are here necessary. As a finite region of the bow is 
in contact with the string, it is not possible to specify the position 



* The principal Helmholtziau type is the mode of vibration in which the time- 
displacement graph of every point on the string is a simple two-step zig-zag. 



24 C. V. RAMAN. 

of the bow by a single constant. Accordingly, the positions of the 
inner and outer edges of the region of contact were noted in the 
observations. The graph therefore shows two curves connecting 
the positions of the two edges of the bowed region with the 




l * 34 66 7 CMS 

DISTANCE OF BOWED POINT FROM BRIDGE. 

( LENGTH OF STRING 33-6 CMS.} 
FIG. i. 

magnitude of the bowing pressure, The ordinates of the graphs 
represent the values of the minimum bowing pressure found neces- 
sary to elicit a full steady tone with pronounced fundamental. 
[For bowing pressures smaller than this minimum, the fundamental 



Mechanically-Played Violins. 25 

falls off in intensity, and the prominent partial becomes its octave 
or twelfth. In certain cases, as for example near the wolf-note 
pitch, we get f cyclical' or * beating' tones.] It will be noticed 
from the graphs that the bowing pressure necessary increases with 
great rapidity when the bow is brought near the bridge. 

The curve in Fig. i (heavily drawn) lying between the ex- 
perimental graphs is a representation of the algebraic curve # 2 y= 
constant. (The constant was, of course, suitably chosen.) It 
will be noticed that the graph follows the trend of the experimental 
values quite closely. In other words, we may say that in the 
cases studied, the bowing pressure necessary varies practically in 
inverse proportion to the square of the distance of the bow from 
the bridge. It may be readily shown that this is the result to be 
expected from theory. Iti my monograph,* I have shown that the 
minimum bowing pressure P is given by the formula 

P ' V 

p= j| ' 



where P A ' is the maximum value at any epoch of the series 

sin (^ + ^,,'J 



sm 



and P , is the value of the series at the epoch at which the bowed 
region of the string slips past the bow. /* is the statical coefficient 
of friction, and H A is the dynamical coefficient of friction during the 
epoch of slipping. B i9 B. ly etc. are the amplitudes of the partial 
vibrations of the string, k {y &. z , etc. are numerical constants for 
the respective partials depending on the instrument, the mass, 
length and tension of the string, and # is the distance of the bowed 
point from the end of the string. We have already seen that ampli- 
tudes B n of the first few partials for a given speed of bowing vary in 
inverse proportion to the distance of the bow from the bridge, and 
their relative phases remain unaltered. In respect of these partials, 
the factor 

, . 

i/sm 



* Bulletin No. 15, pages 73 to 75. 



26 C. V. RAMAN. 

also varies practically in inverse proportion to X Q , so long it is a 
small fraction of /. 

To effect a simplification, we may proceed by ignoring the 
influence of all the partial vibrations except the first few, an 
assumption which is justifiable in the case under consideration, 
as by far the greater proportion of the energy of violin-tone is 
confined to the first few partials. Further, we may for sim- 
plicity, treat the difference (/* H A ) between the statical and dy- 
namical coefficients of friction as practically a constant quantity. 
This will not introduce serious error, provided the speed of the bow 
is not very small. For, if the slipping speed be fairly large, any 
changes in it due to change of the position of the bowed point 
would not seriously alter the dynamical coefficient of friction. On 
these simplifying assumptions, it will be seen from the formulae 
given above that, within the limits considered, the minimum bow- 
ing pressure should vary in inverse proportion to the square of the 
distance of the bow from the bridge, exactly as found in ex- 
periment. This relation would, of course, cease to be valid when the 
bowed point is removed too far from the bridge or when the speed 
of the bow is very small. 



SECTION IV. RELATION BETWEEN BOWING SPEED AND BOWING 

PRESSURE. 

The changing of the speed of the bow is another of the well- 
known resources of the violinist. The principal effect of this is to 
alter the intensity of tone. Pari pass^i with the change of speed 
of the bow, other things remaining the same, the violinist has to 
alter the pressure of the bow. The relation between these may be 
readily investigated with the mechanical player. The experimen- 
tal results for the D-string and for a particular position of the 
bowed point are shown in Fig. 2. 

The graph shows the following features : (i) for very small bow- 
ing speeds, the bowing pressure tends to a finite minimum value ; 
(2) the increase of bowing pressure with speed is at first rather 
slow ; (3) later, it is more rapid, the pressure necessary increasing 
roughly in proportion to speed, and for large amplitudes of vibra- 
tion possibly even more than in proportion to the speed of the 
bow. 



Mechanically -Played Violins. 27 

The foregoing results are, broadly speaking, in agreement 
with what might be expected on theoretical grounds.* This can be 
seen from the formula for the bowing pressure referred to in Section 
III. With increasing speed of the bow, the amplitudes B n of the par- 
tial vibrations increase in proportion, so that if the difference M~^ 
between the statical and dynamical coefficients of friction be 



H 



7 



5 IO 15 20 Sift 30 

VELOCITY OP BOW CMS ./SEC. 

FIG. 2. 



regarded as a constant, the bowing pressure necessary should vary 
directly as the speed of the bow. For very small speeds of the bow, 
however, it is not correct to take **-/*.! as constant, and it would 
be nearer the mark for such speeds to take /*-/" as proportional 
to the velocity of slip, that is, as proportional to the speed of the 
bow. Thus, for very small speeds of the bow, the pressure neces- 

* Bulletin No. 15, pages 151-153. 



28 C. V. RAMAN. 

sary should be nearly independent of the speed, that is, should 
converge to a finite minimum speed. For larger speeds of the bow, 
it would be correct to take /* p A as constant and the bowing 
pressure should then vary proportionately with the speed. For 
very large speeds, the theory of small oscillations would no longer 
be applicable, and the quantities k l9 k 2) k 3 , etc. might increase 
with the speed of the bow. For such large speeds, the bowing 
pressure necessary might increase more than in proportion to the 
speed of the bow. 

A more precise discussion of the experimental results would be 
possible on the basis of quantitative data as to the manner in 
which the coefficient of friction between rosined horse- hair and 
catgut varies with the velocity of slip at different pressures. 

SECTION V. VARIATION OF BOWING PRESSURE WITH PITCH. 

The pitch of violin tone depends on (i) the linear density of 
the bowed string, (2) its length, and (3) its tension, and may be 
varied by varying any one or other of these factors. In practice, 
the violinist varies the pitch by (i) altering the vibrating length 
by "stopping" down the string on the fingerboard, or (2) by 
passing from one string to another. The mechanical player may 
be used to investigate the dependence of bowing pressure upon 
pitch when the latter is varied in any of the ways that may be 
suggested. Obviously, the sequence of phenomena observed would 
not be exactly the same for the four strings of the violin as 
these are of different densities and tension, communicate their 
vibrations to the body of the instrument at different points of the 
bridge and also vibrate in considerably different planes relatively 
to the bridge and belly when excited by the bow in the usual way. 
In the experimental work now to be described, a particular string 
of the violin, e.g. the 4th or G-string, was used, and the pitch 
was varied as in the ordinary playing of the instrument by 
4 stopping ' the string at different points. This was arranged by 
clamping the string down to the fingerboard, with a light but 
strong brass clamp shaped like an arch which could be put across 
the fingerboard, passed down upon it and then lightly fixed to it 
by two set-screws at the two ends. The inner face of the clamp 
was lined with leather to imitate the ball of the fingers of the 
violinist and to prevent damage to the strings. 



Mechanically-Played Violins. 



29 



A few remarks are here necessary. In actual practice, when 
the violinist stops down the string so as to elicit a note of higher 
pitch, he generally takes the bow up rather nearer the bridge so 
as to preserve the relationship between the vibrating length of the 
string and the distance of the bow from the bridge. Strictly 
speaking, this should also have been done in the present investi- 
gation. But as it would have been somewhat troublesome and 
involved the risk of errors in the adjustment of the position of the 
bow, it was decided to keep the bow in a fixed position somewhat 
close to the bridge and to find the relationship between the bowing 
pressure and the pitch of the string under these conditions. We 
have already seen that when the bow is fairly close to the bridge, 




270 3fiO 37O 4-2O 4>7O 52O 570 62O 



PIG. 3. Relation between Bowing Pressure and Pitch (without Mute). 

the bowing pressure necessary varies practically in inverse pro- 
portion to the square of the distance of the bow from the bridge. 
Accordingly, the effect of keeping the bow in a fixed position when 
the pitch is altered, instead of it bringing it nearer the bridge at 
each stage, is to decrease the bowing pressure necessary in a 
progressive and calculable ratio. This effect does not accordingly 
interfere with our observation of the characteristic changes of 
bowing pressure with pitch, which are connected with the changes 
in the forced vibration of the bridge and belly of the violin 
brought about by the change in the frequency of excitation. 

The graph in Fig. 3 represents the relationship between 
bowing pressure and pitch within a part of the range of tone 



30 C. V. RAMAN. 

of the violin which includes the first three of the natural frequen- 
cies of vibration of the body of the instrument. The particu- 
lar violin used was of German make, marked copy of Antonius 
Straduarius, the bridge being of the usual Straduarius model. 
The experiments were made on the 4th or G-string, stopped 
down to various pitches. It will be noticed that the graph 
for the bowing pressure shows pronounced maxima and minima. 
There is a strong maximum at 270, another maximum between 
470 and 520, and a distinct hump between 520 and 570. These 
maxima pretty nearly coincide in pitch with the first three 
maxima of intensity of the fundamental in the tone of the violin as 
estimated by ear, in other words with the frequencies of maximum 
resonance of the instrument to the gravest component of the force 
exerted on it by the vibrating string. The maximum lying 
between 470 and 520 is specially interesting as this region exhibits 
the well-known phenomenon of the ' wolf- note/ In the ascend- 
ing part of this portion of the graph, and especially at and near 
the peak of the curve, it is found that when the pressure of the 
bow is less than the minimum required to elicit a steady tone 
with a well-sustained fundamental component, we get t cyclical ' 
or beating tones of the kind described and illustrated by me in 
previous papers.* The rapidity of the beats depends on the pitch 
of the tone which it is attempted to elicit, and also on the pressure 
and speed of the bow. A similar tendency to production of a 
4 wolf-note ' though not so striking, is also manifested in the 
part of the graph between 520 and 570. The maximum in the 
region of 250 to 285 does not show a similar tendency, at any rate 
to any appreciable extent. It would appear that the gravest 
resonance of the violin chiefly involves a vigorous oscillation of the 
air within the belly of the instrument, but not so vigorous an oscilla- 
tion of the bridge and belly as in the second and third natural modes 
of vibration which show the wolf-note phenomenon. Further 
evidence on this point is furnished by experiments on the effect of 
putting a load or mute on the bridge of the violin as will be referred 
to in the following section. 

The formula for the bowing pressure quoted on page 25 
enables the variation of bowing pressure shown in Fig. 3 to be 
explained. In the series 



* Bulletin No. 15, also Phil. Mag. Oct. 1916. 



Mechanically-Played Violins. 31 



K.B. 



. n-irx 

sm -r 



of which the graph practically determines the bowing pressure 
required, the #,/s stand for the amplitudes of the different partial 
vibrations of the string, and the K n 's are quantities which are 
practically proportional to the corresponding partial components 
of the forced vibration of the bridge transverse to the string at 
its extremity. In the case of a string bowed near the end, 



o 
B n / sin - varies nearly as i/n 8 and thus decreases rapidly as n 

increases. Further, in the part of the range of violin-tone covered 
by the graph in Fig. 3, the fundamental component of the vibra- 
tion of the bridge should obviously be well marked, and K { would 
therefore be of the same order of quantities as K 2 , K SJ etc., or 
even larger. Hence the value of the series given above would 
be principally determined by its leading term proportional to /,, 
and the variation of bowing pressure with pitch would practically 
follow the fluctuations of K ly in other words would follow the varia- 
tions in the amplitude of the fundamental component in the forced 
vibrations of the bridge and pass through a series of maximum 
values at the successive frequencies of resonance of the instrument. 
This is practically what is shown by the experimental results for the 
bowing pressure appearing in Fig. 3, and the graph gives us an idea 
of the sharpness of the resonance of the instrument at each of the 
frequencies referred to. It must be remembered, however, that 
the bowing pressure required is also influenced by the terms pro- 
portional to K a , / 3 , etc., that is by the resonance of the instru- 
ment to the second, third and higher partial components of the 
vibration, and some evidence of this also appears in the graph in 
Fig. 3. For instance, though the first resonance of the instrument 
was actually found to be at a pitch of 284, the peak of the curve 
for the bowing pressure is at about 270 as can be seen from Fig. 3. 
This appears to be a consequence of the fact that the course of 
the curve is to some extent modified by the resonance of the octave. 
It is obvious that corresponding to the resonance of the instrument 
to the fundamental tone in the range of pitch from 470 to 570, 
the octave should be strongly reinforced when the pitch lies 



32 C. V. RAMAN. 

within the range 235 to 285 ; hence the peak of the curve for the 
bowing pressure instead of being at 284 is actually shifted towards 
a lower frequency (270) as is seen from Fig, 3. 

Obviously, the experiments described in this section may be 
extended in various directions. The curves for the three other 
strings of the violin, and especially over the whole of the possible 
range of pitch of the tone of the instrument, and the differences 
between the curves obtained with the different strings would de- 
serve investigation. The differences between different violins 
could obviously be studied by this method, and the constants 
K { KI, etc., for any particular violin and string and for various 
pitches may be found experimentally by study of the free and 
forced oscillations of the bridge, and used for a theoretical 
calculation and comparison with experiment of the bowing pres- 
sure required for exciting the tone of the instrument. 

SECTION VI. EFFECT OF MUTING ON BOWING PRESSURE. 

Perhaps the best illustration of the close relation existing 
between the forces required to be exerted by the bow and the 




220 S70 820 57O 4-2O 47O 52O 570 620 



. 4. Relation between Bowing Pressure and Pitch (with Mute). 



communication of vibrations from the string to the bridge and 
belly of the instrument and thence to the air, is furnished by the 



Mechanically-Played Violins. 33 

effect of putting a mute on the bridge on the bowing pressure 
required for eliciting a musical tone. Very striking results on this 
point may be obtained with the aid of the mechanical player. 
Fig. 4 illustrates the relation between bowing pitch and bowing 
pressure observed experimentally, the G-string of the violin 
being used as in Fig. 3, and the results shown in Fig. 3 and Fig. 4 
being obtained under the same conditions except that in the latter 
case, a brass mute weighing 12*4 grammes was clamped to the 
bridge, while in the former case, the bridge was unmuted. The 
great difference between the two cases is obvious, and the change 
in the form of the graph for the bowing pressure shows a very close 
analogy with the change in the character of the forced vibration 
of the instrument and the intensity of the tone of the violin over 
the whole range of pitch produced by application of the mute. 

In view of the discussion of the graph for the bowing pressure 
contained in the preceding section, and the theoretical treatment 
of the effect of muting already given by me in previous papers * 
it is perhaps not necessary to enter here into a detailed examina- 
tion of the subject, and it may suffice briefly to draw attention 
to some of the features appearing in the graph in Fig. 4. It will 
be noticed that there is a peak in the curve at a frequency of 
about 280. This is the pitch of the first resonance of the instru- 
ment, the fundamental component of the vibration being re-inforced, 
and in this case, the position of the peak in the curve for the 
bowing pressure is not appreciably influenced by the resonance of 
the octave as in Fig. 3. It is clear that the pitch of the first 
resonance of the instrument is hardly influenced at all by the 
application of a load of 12*4 grammes to the bridge, and the 
bowing pressure required at the first peak of the curve is nearly 
the same as in Fig. 3. The first natural mode of vibration of the 
violin does not therefore appear to involve any very large vibration 
of the bridge. Following the peak at 280, we have in fig. 4 a very 
high peak near 330 which is the pitch of the ' wolf-note ' as 
lowered by the mute of 12*4 grammes. The great lowering of 
pitch (from 490 to 330) shows that the second mode of vibration 
of the body of the violin involves a very large vibration of the 



* Phil. Mag. October 1916, Nature October 1917, Phil. Mag. June 1918, and 
J Julie tin No. 15, pages 143 to 151. 



34 C. V. RAMAN. 

bridge, and the enormous increase of bowing pressure at the peak 
of the curve is also noteworthy. This can no doubt be explained 
on dynamical principles as due to the very greatly increased 
amplitude of the forced vibration of the bridge due to the loading. 
At higher pitches, the bowing pressure necessary falls off very 
rapidly, though one or two minor maxima (due to the resonance of 
the instrument in its higher modes as altered by the loading) are 
also obtained. The tone of the instrument in the higher ranges 
of pitch when muted is extremely feeble. 

Further investigations which are worthy of being carried out 
would be the study of the effect of gradually increasing the mass 
of the mute on the graph for the bowing pressure, and also of put- 
ting the mute at different places on the bridge. In view of what has 
been stated above, it is clear that we may expect the changes in the 
form of the graph to follow closely the changes in the pitch of 
resonance of the instrument in its various modes produced by the 
loading. 

SECTION VII. OTHER APPLICATIONS OF THE MECHANICAL 

PLAYER. 

The investigations described in the preceding sections may be 
extended in various directions. Some indications have already 
been given on these points, and it will suffice here to suggest some 
of the other possible applications of the mechanical player. As 
the instrument affords a means of bowing the violin at precisely 
measurable speeds and pressures, it furnishes a means by which 
the intensity of violin-tone and its variations with pitch may 
be quantitatively determined and compared with the indications 
of mathematical theory. Various questions, such as for instance 
the effect of heavier or lighter stringing, the effect of varying the 
pressure, speed, and the width of the region of contact of the 
bow, and the position of the bowed region on the tone-quality of 
the violin may be quantitatively studied with a degree of accuracy 
that cannot be approached in manual playing with its undeter- 
mined conditions. Further, the study of the tone-intensity, of the 
bowing pressure curves, and of the vibration-curves of bridge and 
belly of the instrument under quantitative conditions made 
possible by mechanical playing may be expected speedily to clear 
up various structural problems relating to the construction of the 



Mechanically-Played Violins. 35 

violin, e.g. the effect of the peculiar form of the Stradinarius 
bridge, the influence of its position, the function of the sound-post 
and base-bar, the shape of the air-holes, the thickness, curvature 
and shape of the elastic plates composing the violin, and the in- 
fluence of various kinds of varnish. The dynamical specification 
of the constants determining the tone-quality of any violin over 
the whole range of pitch may be regarded as one of the aims 
towards which these investigations are directed. 

SECTION VIII. SYNOPSIS. 

The paper describes the construction of a mechanical violin- 
player intended for study of the acoustics of the instrument, and 
some of the investigations in which it has been applied. The 
principal feature in the player which is worthy of notice is that 
the conditions obtaining in ordinary musical practice are imitated 
with all the fidelity possible in mechanical playing, and the re- 
sults obtained with it may therefore be confidently regarded as 
applicable under the ordinary conditions of manual playing. 
The following is a summary of the results obtained in the four 
investigations described in the present paper: (i) Effect of the 
position of the bowed region on the bowing pressure : it is shown 
that provided the speed of the bow is not too small, the bowing 
pressure necessary within the ordinary musical range of bowing 
varies inversely as the square of the distance of the bow from the 
bridge. (2) Relation between bowing speed and bowing pressure: 
it is shown that for very small bowing speeds, the bowing pres- 
sure necessary tends to a finite minimum value, and the increase 
of bowing pressure with speed is at first rather slow, but later 
becomes more rapid. (3) Variation of bowing pressure with pitch : 
the graph for the bowing pressure for different frequencies shows a 
series of maxima which approximately coincide in position with 
the frequencies of resonance of the instrument. (4) Effect of 
muting on bowing pressure : it is found that the mute produces 
profound alterations in the form of the graph. The bowing pres- 
sure necessary is increased in the lower parts of the scale and 
decreased in the higher parts of the scale. The peaks in the graph 
shift towards the lower frequencies in consequence of the alteration 
in the natural frequencies of resonance of the violin produced by 
the loading, and the change in the form of the graph is closely 



36 C. V. RAMAN: Mechanically-Played Violins. 

analogous to the change of the intensity of the fundamental tone 
of the instrument produced by the muting. 

Some further possible applications of the mechanical player 
are also indicated in the paper. 



HI* Mechanical Illustration of the Theory of 

Large Oscillations and of Combinational 

Tones* 



By Bhabo Nath Banerji, M.Sc., Assistant Professor of Physics 
in the Calcutta University* 



(Plate III.) 

CONTENTS. 

SECTION I. Introduction. 
SECTION II. Description of Apparatus. 
SECTION III. Single Forcing. 
SECTION IV. Double Forcing. 
SECTION V. Synopsis. 

SECTION I. INTRODUCTION. 

It is well known that the principle of superposition of small 
motions is valid only for systems with infinitesimal amplitudes of 
vibration, and that the cases in which it fails owing to the finite- 
ness of the amplitudes may often rise to considerable importance 
in acoustics. Referring to the principle of superposition Helmholtz 
remarks : " There are certain phenomena which result from the 
fact that this law does not hold with perfect exactness for vibra- 
tions of elastic bodies which though almost always very 
small are far from being infinitesimally small. For infinitesimally 
small motions, the moving forces excited by the mutual displace- 
ment of the particles of the oscillating medium are simply propor- 
tional to these displacements, but as soon as these vibrations 
become large the higher powers of the displacements begin to 



38 B. N. BANERJI. 

sensibly influence the motion." On this basis, Helmholtz has 
founded an explanation of the presence in certain cases of 
harmonics in the tones emitted by simple vibrators and of the 
production of combinational tones within the human ear. 

The experimental illustration of Helmholtz 's theory is a 
matter of great interest, and has attracted the attention of many 
investigators. Amongst the most recent work on the subject may 
be specially mentioned, the contributions of E. Waetzmann and 
W. Moser.* The production of combinational vibrations of strings 
by the simultaneous action of two simple harmonic forces varying 
the tension has also been described and illustrated by Professor 
C. V. Raman. f The present work taken up at the suggestion of 
Professor C. V. Raman proceeds on entirely different lines and is 
intended to furnish simple experimental and visible illustrations 
of the theory of large oscillations and of the production of combina- 
tional tones. 

SECTION II. DESCRIPTION OF APPARATUS. 

The apparatus used was originally designed with an entirely 
different object,, but has proved very well suited for the present 
purpose. It was at first intended to be used as an inharmonic 
synthesiser. The construction will be clear from Figs, (i) and (2) 
in Plate III. A wooden swinging arm is pivoted round an axle 
carrying ball-bearings so as to minimise friction (a bicycle-hub 
serves the purpose admirably). The axle is vertically fixed to a 
solid table, so that the swinging arm is free to move in a horizontal 
plane. (See Fig. i in Plate III.) The motion of the arm is checked 
and controlled by two strong steel springs tightly connected 
to it on either side, the other two ends of the springs being con- 
nected (subject to adjustment of tension) with two supports fixed 
to the table. The swinging arm carries a number of hooks from 
which a row of pendulums can be suspended (each by a pair of 
strings from consecutive hooks). When the pendulums are allowed 
to oscillate, they singly or jointly force a small oscillation of the 



* Science Abstracts, Sec. A, May 1919. 

t Bulletin of the Association, No. if, 1914* and Physical Review, January, 



BHABONATH BANERJI 



PLATE 111 




Fig. 2 




CO 

o 

CO 

c 
2i 

E 

o 

u 

o 
co 
to 

c 
o 



u 
to 

o 

CD 

op 

03 

s 

j>> 
o 

JZ. 



c 
o 



13 
-> 
CO 



Q. 
Q. 



Illustration of the Theory of Combinational*. 39 

swinging arm. An arrangement is provided by which the oscil- 
lation of the arm may be magnified and recorded on paper if so 
desired. (This is not an essential part of the apparatus and may 
be left disconnected unless the records are required.) For this pur- 
pose the arm carries a vertical projecting pin which by working 
in a slot carried at the end of a long aluminium pen controls its 
movements. (See Fig. 2 in Plate III.) This pen is pivoted verti- 
cally at a small distance from the pin, the distance between pin and 
pivot being capable of adjustment with the aid of a screw. The 
pen carries a small reservoir of ink at the end the flow of ink 
from the nib-point being made continuous by connecting the reser- 
voir to the midpoint by a piece of cotton. This nib-point rests on 
a drum which can revolve about a horizontal axis by a clockwork 
arrangement and is covered by a sheet of paper. 

The swinging arm in the apparatus as described above will be 
referred to as the symmetric vibrator in what follows symmetric 
because the force of restitution for any displacement on either side 
of the position of equilibrium is the same. The swinging arm as 
controlled by the spring has a natural frequency of its own, its 
vibration when started being rather rapidly damped out on account 
of friction in the ball-bearings. 

In order to make the vibrator asymmetric when desired, the 
following mode of attachment was resorted to. In addition to the 
two steel springs, two steel wires meeting at a very obtuse angle 
control the movement of the swinging arm. (See Fig. 2 in Plate 
III.) This attachment causes the force of restitution acting on 
the arm to be unsymmetrical. The wires tend strongly to resist 
stretching and consequently the restoring force on one side of the 
equilibrium position for the same displacement is not the same as 
that on the other side. 

The vibrator in either case (symmetric or asymmetric) has a 
natural frequency determined by its inertia and the strength of the 
springs, A pendulum hanging from the vibrator when set in 
periodic oscillation exerts a periodic force on the latter the magni- 
tude of the force depending on the weight of the pendulum bob 
and its maximum angular displacement, and the period of the force 
is the same as the period of oscillation of the pendulum. If two or 
more simple harmonic forces of different magnitudes and periods be 



40 B. N. BANERJI. 

required to be superposed on the vibrator, the same number of 
pendulums (which may be called the forcing pendulums) should 
be used, the magnitude being adjusted by the weight of the bob 
and the period by the length of supension. Moreover if two 
forces at different phases are to be superposed on the vibrator, 
this can be done by timing the starting of the two forcing pendu- 
lums to correspond to the required phase difference. The swing- 
ing arm in all cases will vibrate under the joint action of all 
the forces acting simultaneously upon it. This mode of vibra- 
tion of the vibrator can be recorded by the movement of the inking 
pen on the horizontally rotating drum covered with white paper, 
the ink traces showing the time-displacement curve of the vibrator. 

From the vibrator an exploring pendulum can be suspended 
and will then act as a resonator, for if in the resultant oscillation 
of the vibrator a simple harmonic component synchronous with 
the periodic time of the exploring pendulum be present, the latter 
will be set in vigorous oscillation. The sharpness of this resonance- 
will depend on damping and hence the exploring pendulum should 
be fairly heavy. This pendulum while showing its resonance effect 
reacts on the swinging arm being connected to the latter in the 
same way as the forcing pendulums. This mode of vibration of 
the swinging arm in the presence of the resonators will be illus- 
trated in the course of the paper. 

It will be noticed that the pendulum method of illustration 
bears an analogy to the recent work of Professor E. H. Barton 
and Miss Browning on forced oscillations described in the " Philo- 
sophical Magazine'* (1918 and 1919). 

SECTION III. SINGLE FORCING. 

I started a forcing pendulum with a small amplitude and 
recorded the ink trace of the vibrator. 

FIG - (3>- A/VVVVVVVVX 

FIG. (3) represents the vibration of the symmetric vibrator 
when a simple harmonic force of small intensity acts on it. The 
force is small, for the angular motion of the forcing pendulum on 



Illustration of the Theory of Combinationals. 41 

which the former depends is very small. Here the curve is a 
simple sine curve; the motion being small, the first power of the 
displacement determines it. 

FIG. (4) represents the vibration of the asymmetric vibrator 
under the action of simple harmonic force of small intensity. The 
asymmetry in the vibration is shown by the long flat portion of 
the curve which corresponds to very small movement of the 
vibrator against the wire attachment on one side, whereas on the 
other side the amplitude is greater though the curve is nearly flat 
in consequence of the higher powers of displacement determining 
the motion. 

I then started the forcing pendulum with a large amplitude 
and obtained the following ink traces of the vibrator. 



) - /V/VVV\/\A/VV\ 



FIG - 



FIG. (5) represents the large oscillation of the pendulum as 
perfectly traced by the symmetric vibrator. The curve is not a 
simple sine curve and unlike fig. (3) the crests and troughs are 
flattened out, the remaining portions becoming comparatively 
steeper. The bhape of the curve indicates the complexity of the 
vibration caused by the presence of the higher harmonic components 
in the resultant vibration, and the perfect symmetry on both sides 
indicates the absence of the even harmonics. 

FIG. (6) represents the vibration of the asymmetric vibrator 
under the large amplitude of the forcing pendulum. It is very 
nearly the same as fig. (4) except that the asymmetry is more 
marked, and the nearly flat tops of the former curves have become 
flatter still, this portion being traced in the same circumstances 
as fig. (5). 

For an analysis of the above vibrations, I started the forcing 
pendulum with a considerable amplitude and determined its period 
by means of a stop watch. I then made the length of the explor- 
ing pendulum to start with a little longer than that of the forcing 
pendulum and observed the effect of the latter on the former for 
all lengths right up to the shortest possible. 



42 B. N. BANKRJI. 

With the symmetrical vibrator it is found that the lengths of 
the exploring pendulum for which there is maximum resonance 
correspond with periodic times which are respectively the same 
and as one-third the period of the forcing pendulum. For all inter- 
mediate lengths the exploring pendulum takes up no regular 
oscillation except for lengths which are nearly equal to the resona- 
ting lengths. In the latter cases it is observed that the exploring 
pendulum takes up a periodic oscillation with varying amplitudes 
corresponding to beats, the frequency of which diminishes as the re- 
sonating length is approached. This phenomenon of beats helps the 
final adjustment of pendulum length corresponding to a resonance 
frequency. The periodic time for the latter being determined after 
every such adjustment confirms the presence of the fundamental as 
well as the third harmonic. It is observed that the length of the 
exploring pendulum resonating to the fundamental is longer than 
the forcing pendulum. The explanation of this lies in the fact that 
the periodic time of the pendulum with a large amplitude corresponds 
to a slightly longer pendulum with a smaller amplitude. The angle 
of swing of the resonant pendulum is much greater for the third 
harmonic than for the fundamental. It is also observed that the 
exploring pendulum set to give the octave shows no resonance. 
When the forcing pendulum is started with a lesser amplitude there 
is a very considerable decrease in the intensity of resonance, the 
third harmonic dying out more rapidly than the fundamental. 

The observations described above are fully explained when 
we consider the nature of the forces exerted on the vibrator by 
the swinging pendulum. For small oscillations, the pendulum 
exerts a simple harmonic force synchronous with its ow r n k oscilla- 
tion on the swinging arm from which it is suspended. But when 
the oscillation is large, this reaction includes also a third harmonic 
as may be readily shown. 

The differential equation of motion of pendulum is 

d*V 
Ml jp= - Mg sin 

If a be the semivertical angle of swing of the pendulum the 
iirst approximate solution gives 

B = a cos wt , where w / 



Illustration of the Theory of Combinational*. 43 

the second approximate solution becomes 

a' 8 

= (/COS W't - - COS 3 W't 
192 

where w' = w(i- -j-} and a is slightly different from a. 

The horizontal reaction on the vibrator which is 

MS cos sin or Mg ( 0-f 8 ) 

approximately becomes after substitution 

Mg | a' cos z#7 - J a' 8 cos 3 207 j* approximate!}". 

If n be the natural frequency of the vibrator, its oscillation 
will be represented by 

y + n*y = a cos ?#7 - i a /8 cos 3 w't 

the constant factors depending on Mg and the inertia of the 
vibrator being omitted from the coefficients of the force terms in 
the equation. The solution then becomes 



C S 3 W 



n w 

U ~~ W ft* Q i 

It will be noticed from the above equations as well as by the 
mechanical analysis that the complexity of the vibration in fig. (5) 
is due to the presence of the third harmonic, the symmetry of the 
curve being explained by the absence of the even harmonics. The 
two resonances obtained are due to the corresponding forces 
exerted on the vibrator by the large oscillations of the forcing 
pendulum. The angular swing of the resonating pendulum corres- 
ponding to the third harmonic is large ; for the forced motion of the 

vibrator having this frequency is proportional to the factor _ , % > 

and is thus great, the natural frequency n of the vibrator being in 
the actual experiment nearly equal to 320' the frequency of the 
resonant pendulum. Then again the intensities of resonance of the 
primary and the third harmonic depend respectively on a! and a /B 
which explains the phenomena that with small values of a' the 
third harmonic becomes very small, being dependent on <*' 8 which 



44 B. N. BANERJI. 

will then be negligibly small. It will be noticed that the explana- 
tion of the presence of the third harmonic in the motion given 
above does not involve any assumption that the springs controlling 
the motion of the vibrator deviate appreciably from Hooke's Law. 
Such a deviation, if it did exist however, would produce results 
practically analogous to those described above. 

With the asymmetric vibrator it is found that the lengths of 
the exploring pendulum for which there is maximum resonance 
corresponds with periodic times which are (i) equal ; (2) one-half ; 
and (3) one-third of the period of the forcing pendulum. The 
angle of swing for the fundamental resonance is small, the length 
of the exploring pendulum being slightly greater than that of the 
forcing pendulum as remarked before. The second and third har- 
monics decrease in intensity when the amplitude of the forcing 
pendulum is decreased. In the case of the third harmonic the 
decrease is very marked, but the second harmonic persists even 
with diminished amplitudes. The presence of the third harmonic 
needs no further explanation in view of what has been stated above. 
In forming the approximate differential equation of motion for the 
vibrator for the present case, the force corresponding to the third 
harmonic may be omitted for simplicity. The equation then becomes 

y + n*y + #y* = a cos w t 
where ft is the constant of asymmetry. 
A solution to the above is 



f) s 

1 COS Wt T-r^ r ( -T 

w* 2 (^ 4 w j \ n i 



The asymmetric factor /? is involved in the amplitude 
coefficient of the second harmonic and hence the persistence 
of the latter though with smaller intensity for lesser forcing 
as represented by the presence of a* in the coefficient. That the 
intensities of resonance depend on the natural frequency of the 
vibrator is also evident from the solution. 

The following curves illustrate the modification of the vibra- 
tion of the vibrator in the presence of the resonator, the vibration 
curves without the resonators having been reproduced before. In 
getting the records, the forcing pendulum is started with a big 
amplitude and the exploring pendulum adjusted to give the partial- 



Illustration of the Theory of Combinational. 45 

lar resonance is allowed to hang undisturbed. This second pendu- 
lum takes up the oscillation by resonance and the records of the 
vibration are then obtained. 





FIG. (7) 

VAAAAAAA/X/X/V 

FIG. (6) 
FIG. (8) 
FIG. (9) 



FIG. (7) represents the vibration of the vibrator in the 
presence of the resonator which corresponds to the third harmonic 
feebly present in the forcing pendulum. It will be noticed that 
the presence of the resonant pendulum results in an enormous 
magnification of the third harmonic in the resultant motion of the 
vibrator. FIGS. (8), (9) and (10) represent the vibrations of the 
asymmetric vibrator fig. (8) is the vibration of the vibrator in the 
presence of a resonator which is the octave of the force acting ; 
fig. (9) is in the presence of a resonator which corresponds to the 
third harmonic ; fig. (10) is in the presence of resonators correspond- 
ing to the octave and the twelfth combined. It is evident from these 
figures that the effect of the resonant pendulums is very greatly to 
increase the amplitudes of the upper partials in the resultant 
vibration. 

SECTION IV. DOUBLE FORCING. 

I next pass on to the case of two periodic forces acting 
on the vibrator simultaneously. For this purpose another pendu- 
lum is suspended from the vibrator just by the side of the 
first forcing pendulum. The two forcing pendulums are then 
vigorously started. With the symmetrical vibrator it is found 
that the exploring pendulum responds to primaries and twelfths 
of each of the forces acting. In fact, one or other of the forces 
acting singly gives all the resonances obtained, all observations 
being quite similar to those noticed before in the case of single 



4 6 



B. N. BANERJI. 



forcing. No resonance is observed when the exploring pendulum 
is adjusted to correspond to either the sum or the difference of 
the frequencies of the two forcings. The vibrator of course in the 
presence of the two forces takes up a vibration as if the algebraic 
sum of the two forces act singly on it. 

A special case of the resultant vibration when the periods of 
the two primary forces acting are nearly equal is represented by 
the following ink trace of the vibrator. The phenomena of beats 
which the algebraic sum of the two components would give, the 
frequency of the beats being the difference of the frequencies of 
the two components, is clearly illustrated in Fig. (n). 




FIG. (n). 

Having next made the vibrator asymmetric I started the 
two forcing pendulums vigorously and obtained resonance of the 
exploring pendulum in the cases shown in the following table. 
The choice of the two forcing pendulums was such that the sum 
and difference of their frequencies were not the same as any of the 
higher harmonics of either of the forces. 



Period and fre- 


Resonating 


pendulum. 


- - - 


- - 


quency of forcing 
pendulums. 


Period. 


_. 
Frequency. 


Relation. 


REMARKS. 


r t - 2 08 


2'OS 


48 


Primary. 


Corresponding to /> 


or 


0-90 


rii 


Primary. 


,, to q 




I -04 


96 


Octave. 


to 2/> 


7*3 = 0-9 










or 


0'45 


2'22 


Octave. 


to 2q 




0*69 


1-44 


Twelfth 


to 3/> 




0*63 


i-59 


Summational. 


> to p -r 




I -00 


63 


Differential. 


top ~ 



Illustration of the Theory of Combinationals. 47 

The differential equation of motion for the above case is 
represented as 

V + M *y + /ty* = / cos pt + g cos qt 

where p and q are the frequencies of the forcing pendulums, / and ;' 
the intensities of the respective forces acting. In the above 
equation the small forces of frequency 3/> and 39 present in the 
corresponding large amplitude forcing pendulum have been left 
out of the equation for simplicity, the present intention being the 
illustration of the theor} r of combinational. Hence a solution to 
the above approximate equation will not contain the corresponding 
terms, though they are actually present. 

The solution of the above equation is of the form 

y = K + A cos pt -f- B cos qt + C cos 2pt + D cos zqt 
+ E cos at + F cos Bt 

where <r= p+q and & = p -q. 

The results shown in the table are thus explained. The com- 
binationals of the first order where o- represents the summational 
and 8 the differential have for their intensity coefficients 




The above equations explain conclusively the facts that are 
actually observed in the experiment. An increase in /? the factor 
of asymmetry increases the combinationals. The alteration in the 
constant of asymmetry is mechanically illustrated by an alteration 
in the diameter or tension of the wires controlling the motion of 
the vibrator. An increase in / or g the magnitude of the primary 
forces acting increases the intensity of the combinationals. This is 
an illustration of the well-known fact that for the production of 
the combinational tones the generating tones must be loud and well 
sustained. Then again the factor ri* - <r* or n f - 8* indicates that 
the proximity of the frequency of the tone to the natural frequency 
of the vibrator increases that tone considerably. With my appa- 
ratus the summational approaches the above condition and hence 
its comparatively stronger resonance than with the differential. 



48 B. N. BANERJI. 

It was also found that the two forcing pendulums with the 
exploring pendulum resonating to a combinational formed a system 
of which any two when vigorously started produced resonant 
oscillation in the third. For if a and b be the frequencies of the 
forcing pendulums and c that corresponding to the summational 
a-{-b=c and 

c - a = b 

c - b = a 

so that if c and a or c and b are started the respective resonances 
will be a or b corresponding to the differentials as the equations 
will show. 

The following curves show the mode of vibration of the 
vibrator in the asymmetric system. 



FIG. (12). 



FIG. (12) is the vibration curve of the vibrator under double 
forcing and in the absence of any resonator. It also illustrates 
the phenomena of asymmetric beats. 



FIG. (13). 

FIG. (13) represents the modified vibration curve of the 
vibrator in the presence of a resonator tuned to the summational. 
It will be noticed that the component motion corresponding to the 
summational frequency becomes markedly more prominent and 
obvious to inspection. 

The asymmetric vibrator together with the resonators may 
roughly be taken as the mechanical representation of the human 
ear, the drum skin being compared to the asymmetric vibrator 
and the resonating chords of the basilar membrane to the explor- 
ing pendulums. From what has been said before, it will be quite 
clear that when two loud sources of sound affect the drum skin of 
the ear, the chords of the basilar membrane corresponding to the 



Illustration of the Theory of Combinational. 49 

harmonics and combinational tones are excited and the corres- 
ponding sensations are produced. 

The vibration of the drum skin is not the simple sum of the 
motion due to the two primary disturbances acting separately but 
considerably modified because of its asymmetric configuration and 
also its connection to the inner vibrating parts of the ear including 
the basilar membrane. We have seen in the foregoing that the 
motion of the asymmetric vibrator under the joint action of two 
periodic disturbances may be greatly modified by the presence 
in connection with it of resonators tuned to the frequency of the 
combinational vibration, the effect being to magnify the amplitude 
of this part of the disturbance. The experiments thus also illus- 
trate Helmholtz's remark that even where the production of the 
combinational tones occurs within the ear, the association of the 
ear with a suitable resonant cavity connected with it may tend to 
reinforce the combinational tones. Further, it is also not impossible 
that a similar reinforcement of the combinational may occur 
within the ear itself in consequence of the connection with the 
drum skin of the internal parts capable of vibration and having 
free periods of their own. The natural frequency of the drum 
skin unlike the mechanical analogue is small in comparison with 
the frequency of the audible disturbances affecting it, and hence 
the stronger production of the differentials than the summationals 
in the human ear unlike the mechanical analogue where the sum- 
mational is the stronger. 

SECTION V. SYNPOSIS. 

The paper gives a short account of experiments illustrating 
the production of harmonics and combinational tones in systems 
having finite amplitudes of vibration. The apparatus consists of a 
swinging arm pivoted round a vertical axle and free to oscillate in 
a horizontal plane; the forces controlling this oscillation may be 
made either symmetric or asymmetric as desired. From the 
swinging arm a number of pendulums are suspended which can be 
set in oscillation. The behaviour of the pendulums can be observed, 
and provision is made by which the motion of the swinging arm 
due to their reactions may be recorded if desired, in the form of 
a time-displacement graph. 



50 B. N. BANERJI. 

The apparatus has been used to demonstrate the following : 

(1) The presence of the third harmonic in the oscillations 
forced by a pendulum swinging through a large amplitude. 

(2) The presence of the second harmonic in the oscillation of 
an asymmetric vibrator subject to a simple- harmonically varying 
force. 

(3) The influence of a resonator of the appropriate frequency 
attached to the vibrator in either case in magnifying those com- 
ponents of its motion. 

(4) The production of beats under double forcing of symmetric 
and asymmetric systems. 

(5) The production of combinational oscillations of an asym- 
metric vibrator under double forcing, and the influence of the free 
period of the oscillator on the magnitude of the components of 
combinational frequencies in its motion. 

(6) Thei nfluence of resonators tuned to the combinational 
frequencies upon the motion of the asymmetric system when con- 
nected with it. 

In conclusion, the author wishes to express his cordial thanks 
to Prof. C. V. Raman for the facilities put at his disposal and for 
constant interest and encouragement in his research. 



IV* Some Phenomena of Laminar Diffraction 
observed with Mica. 



By Phanindra Nath Ghosh, M,A, Lecturer on Optics in the 
University of Calcutta. 



(Plate IV.) 



CONTENTS. 

SECTION I. Introduction. 

SECTION II. Micro-structure of the Striae in Mica. 

SECTION III. Spectrum of the Laminary Diffraction Pattern. 

SECTION IV. Effects observed close to the Striae. 

SECTION V. Intensity, Colour and Polarisation of the I^arge- Angle Diffrac- 
tion. 

SECTION VI. Mathematical Theory of the Phenomena. 

SECTION VII. Synopsis. 

SECTION I . INTRODUCTION. 

In a paper recently contributed to the Proceedings of the 
Royal Society,* the author has described and explained the inter- 
esting phenomena observed when a sheet of mica is examined by 
the Foucault test or in the Toepler ' * Schlieren " apparatus as it 
is otherwise called. Certain lines or " striae" on the surface of 
the mica appear luminous and beautifully coloured, the colour 
depending on the angle at which the mica is held to the light 
incident on it in the apparatus. It was shown that the striae are 
the boundaries between regions of the mica having slightly 
different thicknesses, and it was pointed out in the paper that the 
colour of a stria as observed in the Foucault test is complementary 



* " On the Colours of the Striae in Mica." Proc. Roy. Soc., 1919, Vol. 96, 
pp. 257-266. 



52 P. N. GHOSH. 

to the colour of the central fringe in the laminary diffraction- 
pattern produced by it and observed in rear of the sheet of mica 
when plane waves are allowed to traverse it. Attempts were 
made to reproduce the colours observed in the Poucault test by 
using glass plates with artificially prepared laminar boundaries, 
obtained by etching out a very thin layer over part of the surface 
of the glass with dilute hydrofluoric acid. These attempts were 
not very successful, and this failure was ascribed in the paper to 
a want of sufficient abruptness or sharpness in the diffracting edges 
thus prepared. On the ordinary elementary theory, a sharp laminar 
boundary should give a diffraction-pattern which is more or less 
exactly symmetrical in configuration about the central fringe.* In 
practice, etched glass plates give diffraction-patterns which are 
markedly asymmetrical, f the fringes on one side of the centre being 
much brighter than those on the other. Careful observations 
showed that even in the case of mica, the laminary diffraction fringes 
produced by the striae often showed distinct asymmetry though 
to a much less extent than in the case of etched glass plates. The 
central fringe which is strongly coloured often showed distinctly 
different tints at its two edges and was occasionally even com- 
pletely bifurcated in colour. It was thought that a closer examina- 
tion of these phenomena, and of the nature of the laminar 
boundaries in mica, would be of interest. The results of the 
investigation are presented in this paper. 

SECTION II. MICRO-STRUCTURE OP THE STRIAE. 

The striae appear to the naked eye as fine hair-like lines on 
the surface of the mica when the latter is examined in diffuse 
light. Under the microscope, however, an interesting structure is 
revealed and the striae appear resolved into minute echelons, or 
staircases, the number of steps in the echelon being often consider- 
able. Under direct illumination and moderate powers, the boun- 
daries of the successive steps of the echelon appear in the bright 
field as fine dark lines which when the highest powers of the 

* R. W. Wood, " Physical Optics," 1914 Edition, page 250. The symmetry 
should be exact when the phase-difference on the two sides of the boundary is IT 
or any multiple of *. In other intermediate cases, the pattern is asymmetrical, 
but in a relatively minor degree. 

t See, for instance, the photograph by Wood published in his book, he. cit. 



Some Phenomena of Laminar Diffraction. 53 

microscope are used may either remain visible as such or appear 
still further resolved into two, three, or more fine dark lines. 
Under oblique illumination, we have the opposite effect, the entire 
field of the microscope being dark and the successive edges of the 
echelon appearing as bright lines. It should be noticed that the 
successive edges in a stria do not all appear equally dark when 
it is seen under direct illumination, nor do they all appear equally 
bright when seen under oblique illumination. It is thus clear that 
the successive steps of the echelon generally represent unequal 
changes of thickness in the mica. Further, the steps are also 
generally of unequal width, and the width of a step may even 
vary from point to point along the length of a stria. In some 
cases a single boundary may be seen split up along its length into 
two or even into three boundaries. The width of a stria is a 
variable magnitude. Striae have been observed of which the 
entire width does not exceed 1/400 mm. or even 1/600 mm., i.e., 
about five wave lengths of sodium light, and broad striae have 
been observed which extend over i/2oth or i/i5th part of a mm., 
that is, from one hundred to one hundred and fifty times this 
wave-length. The number of steps is in some cases fifteen or 
twenty but is usually found to be between six and ten, and may 
occasionally be as small as one, two, or three. 

Taking the case of a typical stria whose micro-photograph is 
shown in fig. i in the Plate, we find it has seven steps, the total 
width of the steps being about i/ioo mm., i.e. about 20 wave- 
lengths. 

The widths of the successive steps in this stria as actually 
measured with a micrometer were found to be as shown below : 
1st Position. 

1. 'ooio mm. 

2. -0025 

3. -ooio 

4- -0015 

5- -0028 
6. *0022 

With the Jamin interferometer it was found that the optical 
retardation produced by the mica on the two sides of this stria 
differed by just half a wave-length of sodium light. On the 
assumption that this retardation is equally distributed between the 



2nd Position. 


3rd Position. 


ooio mm. 


0009 mm. 


0022 ,, 


0023 


ooio ,, 


ooii ,, 


0017 


0018 


0024 


0028 


0026 


ooio 



54 P. N. GHOSH. 

successive steps, the difference of retardation at each step is of the 
order of i/i4th of the wave-length of the D lines. But the actual 
thicknesses are of different magnitudes as is evident from the ap- 
pearance of the different boundaries as seen under the microscope, 
and some of the steps probably therefore represent optical retarda- 
tions even less than 1/20 or i/25th part of the wave-length of 
sodium light. It is remarkable that laminar boundaries represent- 
ing such small optical retardations in a transparent plate are 
clearly seen as fine dark lines under direct illumination in the 
microscope. The visibility of the laminar edges is evidently closely 
connected with their power of diffracting light, and presents some 
interesting points of comparison with the question of the visibility 
of particles of much smaller dimensions than the wave-length of 
light in the microscope. It would seem worthwhile on a future 
occasion to examine the matter further and determine by direct 
observation the smallest thicknesses of the laminar boundaries 
which can be obtained in mica and which can be detected in the 
microscope under direct or indirect illumination. 

The appearance of the laminar edges in etched plates of glass 
under the microscope forms a striking contrast with the phenomena 
observed in mica. They appear as blurred and ill-defined bands 
even under low powers, and become altogether indistinguishable 
when objectives of higher powers are used. 

SECTION III. SPECTRUM OF THE LAMINARY DIFFRACTION 

PATTERN. 

As remarked in the introduction, the distribution of intensity 
and colours in the diffraction-patterns produced by the striae in 
mica often shows a distinctly noticeable asymmetry of which the 
explanation is evidently connected with the echelon-like nature of 
the boundary. A delicate method of observation which would 
render this asymmetry strikingly evident and susceptible of 
measurement became necessary and was ultimately found in the 
spectroscopic examination of the diffraction-pattern. When the 
laminar diffraction fringes are allowed to fall crosswise on the slit 
of a spectroscope, the spectrum as seen is found to be crossed by 
dark bands showing the colours obscured by interference in the 
different parts of the pattern. If the distribution of intensity in 
the laminar diffraction-pattern had been strictly symmetrical, we 



PHANINDRA NATH GHOSH. 



PLATE IV. 



Fig. 





Fig. 5 



Fig 2 i 




Fig. 6 



Fig. 3 




Fl ' 4 liilf : " 7 3f"- 




Fig. 7 



Laminar Diffraction phenomena observed with Mica. 



Some Phenomena of Laminar Diffraction. 55 

should evidently see in the spectrum a system of bands with a 
configuration symmetrical about the central line. Actually, how- 
ever, we obtain the very curious appearance of bands running 
obliquely through the spectrum (Figs. 2, 3, 4 in Plate IV).* 
Fig. 2 which shows quite a large number of bands was obtained 
with a rather thick stria which showed merely a grey colour in the 
central fringe. Here every one of the bands is inclined, the inclina- 
tion gradually diminishing as one proceeds towards the violet 
end of the spectrum. The actual shape of the individual bands 
running through the spectrum is an oblique /of which the tips are 
considerably fainter than the middle. The same characteristics 
are also shown by fig. 3 in which we find four oblique bands dis- 
tributed practically equally throughout the spectrum, and the 
distribution of intensity above and below these bands in the 
spectrum is obviously quite different. The phenomena are still 
more clearly seen in fig. 4 which was obtained with a stria which 
gave a green-coloured central fringe. The obliquity of the two 
dark bands running through the spectrum and the asymmetric 
distribution of intensity above and below these bands is particular- 
ly noticeable in this figure. Fig. 5 represent sthe spectrum of the 
fringes due to a stria giving a half wave-length change in optical 
retardation. Here, though the band runs straight through the 
spectrum, a distinct obliquity in its position and an asymmetric 
distribution of intensity on the two sides of it were both noticed. 
The intensity was greater on one side of the fringe than on the 
other, and the fainter bands which accompany the central dark 
band on either side are much clearer and further from it on one 
side than on the other. 

The relative optical retardation on the two sides of a stria 
should obviously be increased by tilting the mica, and it is interest- 
ing to watch the effect of this on the position and number of the 
bands in the spectrum of the laminary diffraction-pattern. The 
observations are best made with a stria which shows a small num- 
ber of bands in the spectrum. It is found that as the mica is 

* It is found that superposed upon these bands, there also appear in the 
spectrum a set of very fine, numerous and rather diffuse bands rnnning parallel to 
the slit of the spectroscope. These are the well-known bands due to the inter- 
ference of the light transmitted through the mica with the light which has passed 
into the spectroscope after two or more internal reflections within the mica, and 
need no further remarks. 



56 P. N. GHOSH. 

turned gradually from the normal to the oblique position with 
respect to the light incident on it, the bands shift towards the red, 
and the number of the bands visible in the spectrum also increases. 
Phenomena which are roughly analogous to the above but not 
so beautifully clear and regular are obtained when the spectrum 
of the laminar diffraction of a very thinly etched glass plate is 
observed. The appearance of asymmetry in this case is consider- 
ably more exaggerated, the fringes on one side of the pattern 
being numerous and showing well-marked contrasts, and the 
fringes on the other side being few in number and hazy in outline. 

SECTION IV. EFFECTS OBSERVED CLOSE TO THE STRIAE. 

The phenomena described in the preceding section are those 
observed at comparatively large distances (of the order of a 
meter) from the mica. It is of interest to examine the effects 
observed near the sheet of mica, say within a few centimeters of it, 
as we may naturally expect that the asymmetry of the pattern 
due to the finite width of the striae would be much more marked 
in their immediate neighbourhood. For the purpose of this study, 
it is found convenient to use striae which run more or less straight. 
The source of light is a slit placed at a sufficient distance from the 
mica and parallel to the direction of the striae, and illuminated by 
sunlight or the light of an electric arc. The diffraction-fringes 
formed in the rear of the striae may be observed through a micro- 
scope. For spectroscopic analysis of the pattern, the eyepiece of 
the microscope may be removed and the image of the diffraction- 
pattern formed by the objective may be allowed to fall on the slit 
of a direct-vision spectroscope or of a constant- deviation wave- 
length spectrometer. By racking out the objective of the micro- 
scope, the complete succession of phenomena commencing from the 
plane of the mica itself right up to any desired distance from it 
may be observed in quick succession, and a vivid idea obtained of 
the whole series of effects. The observations made in this manner, 
especially with the aid of the spectroscope as described above, are 
extremely useful in getting at a clear understanding of the whole 
case. 

The phenomena noticed naturally depend a good deal on the 
particular stria under study, especially on the number, width, and 
height of the steps in the staircase and the total relative optical 



Some Phenomena of Laminar Diffraction. 57 

retardation on the two sides of it. Nevertheless, certain general 
features are observed which are common to most of the cases 
studied. When the focal plane of the objective coincides as nearly 
as possible with the mica, the structure of the stria is clearly seen , 
the successive edges of the echelon appearing as fine dark lines. 
Even at this position, however, a number of very fine (practically 
equidistant) fringes rnay be seen bordering these edges, these being 
more marked on one side of each edge than on the other. The 
simplest case is that in which the echelon consists of but a single 
step. Occasionally striae may be obtained consisting of but a 
single edge not resolvable by a i/8th inch Reichert objective. 
Even in such cases, however, the laminar diffraction-pattern at 
close quarters often shows a distinct and sometimes quite marked 
asymmetry, the fringes on one side being clearer, brighter and 
more numerous than on the other, and the central fringe appearing 
much darker than it is at a distance from the mica. These 
features persist till the focal plane is drawn away from the mica 
to a distance of a centimeter or two. Apart from these special 
characters, however, the pattern due to an echelon of a single 
step is very similar to what we should expect on the elementary 
theory in the case of a perfectly abrupt laminar boundary. 

With striae consisting of two, three or more steps in the stair- 
case, the effects are naturally more complex than in the case of a 
single step. As the focal plane is drawn away from the mica, the 
fringes bordering the successive edges rapidly broaden out and be- 
come superposed on each other. Soon, all trace of the structure of 
the stria is lost owing to this superposition, and the field of view 
may be then seen clearly differentiated into three parts the effects 
observed in which may be separately considered. 

First, the central part of the field which is much darker than 
the rest of the field. The width of this part of the field differs for 
different striae, being large for the broad striae, and small for the 
narrow ones. This central region is seen filled with a succession of 
faint but markedly coloured fringes which are at first narrow and 
numerous, but widen out and become fewer in number as the focal 
plane is drawn away from the mica, till finally only one or two 
coloured bands remain in the middle of the field. 

Secondly, the part of the field lying on the thicker side of the 
stria. This is the brightest part of the field, and contains numer- 



58 P. N. GHOSH. 

ous well-marked fringes which at first are nearly equally spaced 
and extend to a considerable distance from the central part of the 
field. As the focal plane is drawn away from the mica, these 
fringes become broader, less bright, and their spacing becomes 
more unequal. Fewer fringes are also then visible. 

Thirdly, the part of the field on the thinner side. This usually 
contains only a few faint and hazy diffraction fringes. The 
contrast between the second and third parts of the field is very 
marked but decreases gradually as we recede from the mica. 

Corresponding to the foregoing changes in the microscopic 
appearance of the fringes, the spectrum of the diffraction-pattern 
also alters. In the central part of the field, a number of narrow 
horizontal bauds is at first seen in the spectrum. As the objective 
recedes from the mica, these bands widen out, those on the thicker 
side of the mica slide away towards the red end of the spectrum, 
those on the thinner side slide away towards the blue end, and the 
remaining bands gradually assume an oblique position and tend 
to set themselves in the spectrum less inclined to the slit of the 
spectroscope as we recede from the mica. When the focal plane is 
within a centimeter or two of the mica, a large number of very 
dark and bright fringes may be seen in the region of the spectrum 
on the thicker side of the stria, and relatively few and more hazy 
fringes on the thinner side. These fringes in the spectrum gradu- 
ally widen out and become less numerous and more hazy in outline 
as we recede from the mica. 

Figs. 6 and 7 in the Plate illustrate the preceding remarks and 
represent microphotographs (in the light of the electric arc) of the 
diffraction-effects observed close to the striae, fig. 6 having been 
secured with the focal plane nearer the mica, and fig. 7 with the 
focal plane somewhat further away. As the light used was not 
monochromatic, the photographs do not convey an adequate idea 
of the very large number of fringes visible when the pattern is 
analysed by the spectroscope. 

The diffraction-effects due to the laminar boundaries in 
etched glass plates observed in their neighbourhood through a 
microscope are not so striking as those observed with mica, the 
number of fringes visible, specially in the spectral analysis of the 
pattern, being much smaller than with mica. Apart from this, 
however, the effects observed in the two cases are broadly analogous,. 



Some Phenomena of Laminar Diffraction. 59 

the asymmetry being considerably more exaggerated in the case of 
etched glass plates. 

SECTION V. INTENSITY, COLOUR AND POLARIZATION OF THE 

DIFFRACTION. 



On account of the extremely fine structure of the laminar 
edges forming the striae in mica, they possess the property of 
scattering or diffracting light in directions making large angles (up 
to 180) with the light transmitted through or reflected from the 
plane surface of the mica. The feature of this large-angle diffrac- 
tion which is most noteworthy is that the striae as seen by the light 
diffracted by them appear very much more intensely luminous on the 
retarded side of the wave-front and relatively quite feebly lumin- 
ous on the other side. Observation also shows that when white 
unpolarized light is incident on the striae, the light scattered by 
them is often strongly coloured and also polarized. The investiga- 
tion of these effects is a matter of some complexity, as they are 
found to depend on a number of factors, (a) the fine structure of 
the laminar edge and the relative optical retardation on the two 
sides of it, (6) the angle of incidence of light, and (c) the angle of 
diffraction. Further, though the problem of the large-angle 
diffraction of light by the edge of a semi-infinite perfectly reflecting 
screen has been investigated by Sommerfeld and shown to involve 
polarization effects, the corresponding problem of the large-angle 
diffraction by the edge of a thin lamina of transparent solid yet 
remains unsolved, and no theoretical guidance for research in this 
direction is therefore available. It is hoped to investigate these 
effects in detail experimentally when a suitable opportunity occurs. 

SECTION VI. MATHEMATICAL THEORY. 

The observations considered in the preceding sections which 
require explanation are the following : (a) the appearance and 
visibility of the laminar boundaries in the microscope when 
focussed upon them in direct illumination; (b) the diffraction 
phenomena observed close to the striae, and (c) the effects ob- 
served at a distance, especially the obliquity of the bands appearing 
in the spectrum of the diffraction-pattern. The investigation of 
(a) is a question relating to the theory of microscopic vision 



6o 



P. N. GHOSH. 



which the author hopes to be able to deal with in a later paper. 
The investigation of (b) and (c) strictly speaking requires an 
exact knowledge of the structure of the particular stria under 
observation. As our present purpose is, however, merely to get 
a general idea of the explanation of these effects, it is sufficient if 
the stria is assumed to be an approximately wedge-shaped bound- 
ary separating regions of the mica having slightly different thick- 
nesses. Such a wedge-shaped boundary would diffract light in 
roughly (though not by any means in exactly) the same way as 
the equivalent staircase structure. 

Considering now the effects in the neighbourhood of the 




FIG. 8. 



stria, fig. 8 showing the geometrical path of the rays gives us at 
once a general idea of the phenomena to be expected. The 
central relatively dark part of the field, the increased brightness 
and the numerous interference fringes seen in the region on the 
thicker side of the mica, and the relatively few and hazy diffrac- 
tion fringes seen on the thinner side of the mica are all exactly 
what we should expect on the principles of the wave-theory. 
The following is the detailed mathematical treatment : 

The form of the wave front on emergence from the plate shown 
in fig. 8 would be that shown by the upper line in fig. 9. The effect 
at any point O in the field may be readily found in terms of Fresnel's 



Some Phenomena of Laminar Diffraction. 



61 



integrals. In fig. 9, r is the distance of the pole from the point of 
observation and x l , x, 2 the distances of the two edges of the stria 
from the pole, p is the relative retardation of the wave front 
on the two sides of the stria. 



We may write 



FIG. 9. 

We then get as the expression for the amplitude omitting 
constants 



Tutting 



Cos27r -"* 



COS2, -- 



r A 



_ I 

the expression for the amplitude becomes 



62 P. N. GHOSH. 

f 00 / # x f-** 

\ cos2T ( A - ]dx+ \ cos 

-'-A', ^ 2y O X / J-oo 

f"~^ 1 / ^ \ 

+ V cos 2* [ A 0* \dx 

J-, a ^ 2 " ^ 

which may be transformed into the usual form of Fresnel's 
integrals by making 



V = 



We get the expression for amplitude to be 
cos 2?r A 



r<* TOO 

\ cos ^ ^vVw+sin 27T ^ i sin \ -rrv^ 

J z;j / t/i 

/~^ / ^2 

1 cos J 7T7;Vi;+ s i" ^ (^1 - ^) I si 

J 00 J -CO 

r ^3 

l sin | ^ 

J Vi 

>^CO /OO 

a^ == 1 cos | TT^^T;, #. 2 = I sin ^ irw*rfu 

J v\ J Vi 

rv-i /~ V 2 

&i C = \ cos i ^Vy, 6. z =1 sin J irv*dv 

J - oo J 00 

/ 8 / V 3 

c t ==> I cos J Kv*dv, c z = V si 
J 1/4, */ 1; 4 



+ cos 2** (^4 - ^) 1 cos J 7T7;Vi;+ s i" ^ (^1 - ^) I sin 

J 00 

^^3 
4-cos 27r (A - BI) [ cos 

putting 



sn 



We get the following expression for the intensity 

cos 2^+c L cos 2^ a - 6. z sin 27r9 L - c % sin 
^ n 2^+^ sin 2^ 2 +& a cos 2w^+c 4 cos 2^ a )* 

In the particular case of a stria whose spectrum of laminary 
diffraction is illustrated in fig. 4 in the plate, 



Some Phenomena of Laminar Diffraction. 

p = '00054 m/m 

*i ~ *a * 02 5 m / m 
r = i meter 

the illumination curves for three wave lengths 

A = '00050 m/m 
X = 00054 m/m 
X = -00060 m/m 

have been calculated and plotted (fig. 10). 

tog 10 



2-52 ~MT~ 1 -6 0-5 1 -52 2-5^ 

\- ooo f4 Tn/m 







/^^ 
















3 


I 

- mint 




V/ 


/ 




































52 16 1 -5 5 1 15 a 2-5' 



A 




"V 




^^ 


~ 


X 










b 
a 


u_ 
l 




Vx 


/ 








\ / 


/ 






I 
n 














V 









2-5 2 15 1 



\ 1-5 2 



They clearly show that 

(i) for wave length * = -00050 m/m the central dark 
fringe has somewhat shifted to one side ; 



64 P. N. GHOSH. 

(2) for wave length A. = -00054 m M the central dark fring 

has vanished ; 

(3) for wave length * = *ooo6o m/m the central dark fringe 

has shifted to the opposite side. 

This is precisely the effect which appears in fig. 4 in the plate 
on the two sides of the transmission region in the spectrum. 

SECTION VII. SYNOPSIS. 

In a previous paper (published in the Proc. Roy. Soc. for 
November 1919), the author has described and explained the colours 
shown by the striae or laminar boundaries in mica when the same 
is examined by the Foucault test. The present paper describes 
some observations on the micros tructure of these laminar bound- 
aries and of the various diffraction effects produced by them. The 
following are the principal results obtained. 

(a) The stride appear resolved in the microscope into minute 
echelons or staircase structures, the number of the steps varying 
from one to ten or fifteen for different striae. The optical retarda- 
tion due to any particular edge in the echelon is generally quite a 
small fraction of a wave length. The edges are nevertheless 
clearly visible in the microscope as very sharp dark lines. 

(b) In consequence of the structure of the striae above men- 
tioned, the laminar diffraction pattern observed even at a consider- 
able distance from the mica shows distinct evidence of asymmetry 
in the distribution of intensity and colour of the fringes. 

A very delicate method of exhibiting this asymmetry is fur- 
nished by spectroscopic analysis of the laminar diffraction pattern, 
the dark bands due to interference running obliquely through the 
spectrum and being much more clearly marked on one side of the 
pattern than on the other. 

(c) In the immediate neighbourhood of the striae the diffrac- 
tion-phenomena as observed through a microscope are more com- 
plicated, the asymmetry being very marked, and a very large 
number of fringes may be observed. 

(d) The striae scatter light through large angles, the light thus 
diffracted showing a marked asymmetry in its intensity on the two 
sides of the direction of the incident light and also exhibiting both 
colour and polarisation. 



Some Phenomena of Laminar Diffraction. 65 

The investigation described in this paper was carried out in 
the Palit Laboratory of Physics, and the author's best thanks are 
due to Prof. C. V. Raman for his suggestions and unfailing interest 
in the work during its progress. 



V. On the Forced Oscillations of Strings under 
Damping proportional to the Square of the 

Velocity* 



By Rajendra Nath Ghosh, M.Sc., Research Scholar of the 

Association. 



(Plate V.) 



CONTENTS. 

SUCTION I. Introduction. 

SECTION H. I/aw of Damping of Free Vibrations of Strings. 

SECTION III. Experimental Study of Forced Vibrations of Strings. 

SECTION IV. Theory of Forced Oscillations under Transverse Excita- 
tion. 

SECTION V. Theory of Forced Oscillations under Longitudinal Exci- 
tation. 

SECTION VI. Synopsis. 



SECTION I. INTRODUCTION. 

In a recent paper published in the Physical Review * J. Parker 
Van Zandt has given a discussion and comparison with experi- 
ment of the theory of free oscillations of bodies subject to resisting 
forces proportional to the square of the velocity. He has also 
given a very full bibliography of the literature on oscillations sub- 
ject to this and other special laws of damping. In the course of 
some quantitative work on the forced oscillations of strings under 
different types of excitation recently undertaken by the present 
author, results have been obtained which show that the frictional 
forces acting on a stretched string vibrating in air are not propor- 



* Physical Review, Nov. 1917, page 415. 



68 R, N. GHOSH. 

tional to the velocity of each point on the string as is generally 
assumed,* but increase much more rapidly than in proportion to 
the velocity. It accordingly appeared to be of interest to investi- 
gate the theory of forced oscillations under damping proportional 
to the square of the velocity, and to compare the results with 
those found in experiments on stretched strings vibrating under 
different types of excitation. This has been done, and the present 
paper describes the outcome of the investigation. The experimental 
work has been carried out with the aid of the apparatus for the 
study of vibrations of strings developed by Prof. C. V, Raman and 
described in a recent publication. f This apparatus is specially 
suitable for the present investigation and has indeed made it pos- 
sible. 

SECTION II. LAW OF DAMPING OF FREE VIBRATIONS OF A 
STRETCHED STRING. 

Observations have been made by the author of the rate of 
decay of the free oscillations of a stretched cord (of twisted cotton 
thread) vibrating in air. This was investigated by a photographic 
method. The middle point of a stretched string was pulled aside 
and then released. The damped oscillation of the point was 
photographed on a moving plate, and measurements of the ampli- 
tudes were made by means of a cross-slide micrometer. The 
following table shows a typical set of results. In column I we 
have the successive amplitudes, in column II their successive 
differences D, in column III the mean M of two successive 
amplitudes. 



* Ivord Rayleigh, " Theory of Sound," Vol. I, Arts. 131-134. 
t C. V. Raman, " An Experimental Method for the Production of Vibra- 
tions,*' Physical Review, Nov. 1919. 



The Forced Oscillations of Strings. 



69 



TABLE I. 
Length of Siring 100 cms. 



No. 


Successive 
amplitudes. 


D 






cms. 






I 


2-0344 
1-6785 


3559 


i-i 


2 


1-4102 


2683 


I'f 


3 


1-2115 


1982 


i'3 


4 


1-0363 


1752 


1*1 


5 


0-9069 


1294 


o'S 


6 


0-8147 


OQ22 


0-8 


7 


0-7050 


1097 


0-7 


8 


0-6424 


0626 


0-6 



M 



D/M* 



13 



12 



D/M 



19 
17 

is 

15 

13 
II 

'14 
09 



In the fourth column it will be observed that D/M* is almost 
constant whereas D/M shows a variation much greater than the 
experimental errors. Now we know that when the damping is 
proportional to the first power of the velocity, the ratio of two 
successive amplitudes is constant, from which it can be easily 
shown that the difference of two successive amplitudes divided by 
their mean must also be a constant. But in the present case it is 
seen that the above ratio namely D/M is not constant, while D/M* 
is constant. Hence the law of friction al resistance is not that of 
the first power of the velocity. We shall presently show that the 
result D/M* is a constant expresses the fact that the frictional 
forces are proportional to the square of the velocity. 

The equation of motion of a particle on the stretched string 
vibrating freely and resisted by forces proportional to the square 
of the velocity is given by 

#= - n*U kV* (i) 

where U is the displacement and the constants have their usual 
meanings. For a first approximation, neglect the term kU*, 
and we have 

U=sA cos nt (2) 



70 R. N. GHOSH. 

By the method of successive approximation we get the value of U 
when resisting forces are taken into consideration. 

U = A cos nt + \A*k + ;, A*k cos 2 nt (3) 

dU 

Let U***A { and =o when t=o 
at 

Therefore we get A =? A } --$kA* and U is then given by 

U = (A | l/t/1,*) cos nt + /!,*/! + ^A*kcos2nt , . . (4) 

Equation (4) is only true for half an oscillation. At / = *jn the 
displacement U is given by 

C7- -(/Ij- j /!,*) 

Hence the amplitude of swing is diminished by \ A ,* k = > 
(approx.). Hence we arrive at once at the result that 

D/M* - i ft (5) 

From the experimental results we have seen that D/M* is almost 
a constant. Hence we infer that the frictional forces for the large 
amplitudes of vibration employed are proportional to the square 
of the velocity. 

From other results obtained by the author it would seem 
that for very small amplitudes of vibration on the other hand, 
the frictional forces do not increase quite so rapidly as in pro- 
portion to the square of the velocity, and are more nearly pro- 
portional to the first power. 

SECTION III. -EXPERIMENTAL STUDY OF FORCED VIBRATIONS 

OF STRINGS. 

Description of apparatus. 

For an experimental study of the forced oscillations of 
stretched strings, an electric motor-vibrator was used. A detailed 
account of this apparatus as originally devised by Professor J. A. 
Fleming, and subsequently modified and improved by Professor 
C. V. Raman, has been published in the Physical Review* In 
Plate V, Fig. i shows the general features of the apparatus. A 
circular wheel is fixed to the axle of a small motor. To this wheel 
a brass disk carrying a slot and movable pin is fixed. The pin 

* November, 1919. 



RAJENDRA NATH GHOSH. 



PLATE V. 




Fig. 




Fig. 2 



Fig. 3 



Forced Oscillations of Strings. 



The Forced Oscillations of Strings. 71 

actuates a crank shaft and oscillating lever moving between guides. 
The oscillating lever performs what is approximately a simple 
harmonic motion, and excites the vibration of the string in the 
manner of Melde's experiment. The frequency of this excitation 
depends upon the rate of revolution of the motor ; and the 
amplitude of motion of the lever-arm can be governed to a nicety 
by altering the position of the pin on the motor. When the pin 
is exactly at the centre, the amplitude of motion of the lever is 
zero, and as the pin is moved away from the centre, the amplitude 
increases. This adjustability of the amplitude enhances the utility 
of the apparatus. The frequency of the vibrator, i.e. of the lever 
arm, was kept constant by a sliding rheostat and the constancy 
was observed through a stroboscopic tuning fork. In all the 
experiments, the frequency was generally 60 per second. The 
tension was adjusted by putting weights on a pan and clamping 
the end of the string near it, so that the pan could not oscillate. 
The motor-vibrator permits the use of long strings, considerable 
tensions, and gives a perfectly constant amplitude, advantages 
which are not possible when a tuning fork is used as the exciter. 

The amplitude of vibration of the string depends on (l) its 
tension and (2) the magnitude of the obligatory motion imposed 
on it at the end. Keeping (2) constant, (i) may be varied and the 
resonance curve of the string may be traced. This may be done 
for a series of values of (2). 

Transverse Excitation. 

The case first studied in the manner described above was the 
transverse type of Melde's experiment in which the maintained 
oscillation has the same period as the obligatory motion impressed 
at one end. This has been discussed mathematically by the late 
Lord Rayleigh * on the assumption that the frictional resistances 
are proportional to the first power of the velocity. Corresponding 
to an obligatory motion Y=y cos pt at a nodal point, the ampli- 
tude of vibration of the string at any point is approximately 
given by 

4 27T , k X* ft 27T \ \ 

sin* #H -- r cos* x I " 
A_ > 

~ ' ' 



Theory of Sound, V <>1 I, Art i 



R. N. GHOSH. 



so that we should find the amplitude of vibration to be proportional 
to the obligatory motion y. Actually, however, on determining 
the amplitude of vibration of the string for the different values of 




j } u Jltfinitiuk of aUtffatjtrt motia 

2 "" < *a 'Ceo*.'" 

FIG. 4. Relation between Amplitude and Obligatory Motion. 

y, the amplitude increases much less rapidly than in proportion 
to y. The experimental results are shown in fig. 4 in which the 



N 



K 



$s 



\ 



10 20 30 40 5O 60 70 

FIG. 5. Resonance Curves for two different values of y. 

ordinates represent the maximum amplitudes of vibration obtain- 
able with the respective extents of the obligatory motion shown 
as abscissae. The form of the graph is approximately a parabola. 



The Forced Oscillations of Strings. 73 

This result indicates that the f fictional resistance is not propor- 
tional to the first power of the velocity, but, as I shall show later 
on, is more nearly in proportion to the square of the velocity. 

Another important fact noticed was that the tension for 
which the maximum amplitude is obtained decreases with increas- 
ing values of y. This point was studied by adjusting the tension 
of the string, clamping its free end, and then starting the motor 
vibrator. The experimental results are shown in fig. (5) where 
the amplitudes of vibration of the string are plotted against the 
tension for two different values of y. The shift of the maximum 
amplitude and of the whole curve when the amplitude of vibration 
is large, is evident from a comparison of the two curves. The 
theoretical interpretation of this result will be discussed later in 
the course of the paper. 

Longitudinal Excitation. 

The next case studied was the longitudinal type of Melde's 
experiment in which the obligatory motion imposed at one end 
varies the tension of the string. In this case, as has been shown 
by C. V. Raman,* maintenance is possible when the frequencies 
of the obligatory motion and of the oscillation maintained stand 
in any of the ratios 2 : n where n is an integer. In the present 
paper attention will be confined to the first case in which the 
frequency ratio is 2 : i. The theory of this case was first given 
by the late Lord Rayleigh, and later modified and extended by 
Prof. Raman in the papers cited so as to give results more in 
accordance with experiment, the assumption being made that 
the frictional force is proportional to the first power of the 
velocity. The experimental observations of Prof. Raman were 
made using a tuning fork as exciter, and as the reaction of the 
string on the fork materially influences the vibration of the fork 
and alters its amplitude, a difficulty arises in interpreting the results 
observed. With the electric motor- vibrator, on the other hand, the 
amplitude of the obligatory motion is invariable, and its frequency 
can be kept constant by the use of a rheostat so that the reaction 
of the string can exercise no effect. A stricter comparison of the 



* Bulletin No. 6 of the Association, also Phil. Mag., October 1912, and Physical 
Review, December 1912, July 1914 and November 1019 ; see also Jones and Phelps, 
Physical Review^ November 1917. 



74 



R. N. GHOSH. 



experimental results with theory is therefore possible. The experi- 
mental observations may be considered under two heads, (a) 
dependence of amplitude of vibration upon the tension and the 
magnitude of its periodic variation, and (b) dependence of the 
phase of vibration on the same variables. 

(a) : In order to study the* relation between the amplitude 
of vibration of the string and the tension, the latter was made 
very high at the beginning, and then it was continuously dimin- 
ished. With the change in tension, the speed of the motor and 
of the variation of tension produced by it tend to change to a 
slight extent, but they were kept constant as explained above 




pit;, o. Resonance Curve under Variable Spring of Double Frequency. 

with the aid of a sliding rheostat and a stroboscopic fork. It was 
found that at a high tension (much greater than the value 
for which the natural frequency of vibration of the string is half 
that of the vibrator) the amplitude of vibration is small, but has 
always a definite value for a definite tension. As the tension is 
diminished, the amplitude of vibration increases, and continues 
to increase till a stage is reached when the tension is much lower 
than the theoretical value for which the natural frequency is half 
that of the vibrator; the motion of the string then suddenly 
collapses giving place to other types of oscillation. At this in- 
stant the amplitude is a maximum. When the tension is such that 



The Forced Oscillations of Strings. 75 

the natural frequency is half that of the vibrator, the amplitude 
is much less than the maximum value. The amplitude is not the 
same when the tension is greater or less than the theoretical value 
for half frequency by a certain amount. The whole sequence* of 
events is shown in fig 6. The graph shown there has been 
drawn from the experimental data. The continuous increase of 
the amplitude with the diminution of tension, and the consequent 
asymmetric character of the curve aie both clearly shown. The 
largest value of the amplitude obtained at the collapse point is 
shown by one extremity of the curve. 

As the amplitude of the vibrator and consequently also the 
magnitude of the periodic variation of tension are increased, the 
maximum possible amplitude of vibration of the string also 
increases. The experimental curve simply shifts up through a 
certain distance. If on the other hand we diminish the magnitude 
of the imposed periodic variation of tension, the amplitude main- 
tained also diminishes, and when the imposed variation falls below 
a certain very small limit no maintenance is obtained. 

(b) : The phase relation between the exciting force and the 
vibration of the string was easily studied by attaching a bright 
bead to the string near the vibrator. Each point of the string 
has two motions at right angles to each other, the longitudinal 
motion being due to that of the vibrator, and the other the transverse 
motion of the string itself. Hence the Lissajous figure traced by 
the bead gives the phase relation required. Using arc-light to 
illuminate the bead, the Lissajous figures were photographed at 
different amplitudes of vibration of the string. Fig. 2 in Plate V 
shows the figure when the amplitude was small corresponding to 
a high tension. The parabolic figure seen (convex towards the 
vibrator) at once indicates the frequency and phase relations 
between the longitudinal and transverse components of the 
motion, in other words between the maintaining force and the 
maintained vibration. Fig. 3 in the Plate shows the I/issajous 
figure when the amplitude of vibration of the string was large 
corresponding to a low tension. The change in phase which has 
occurred in consequence of decrease of tension is evident. The 
rapidity with which the change occurs depends upon the magni- 
tude of the imposed variation of tension. When this is large, we 
find that the phase relation alters slowly till the form of the 



76 R. N. GHOSH. 

Lissajous figure which is originally a parabola convex to the vibrator 
reaches the limiting form of an 8 curve. At this stage the 
vibration collapses. On the other hand when the magnitude of 
the periodic variation of tension is small, the phase changes rapidly 
with decrease of tension, the initial and final forms of the I^issajous 
figures are the parabola and the 8 curve respectively in the same 
way as in the case previously mentioned. 

SECTION IV. THEORY OF FORCED OSCILLATION UNDER 
TRANSVERSE EXCITATION. 

In this section we shall deal with (i) the forced vibration of 
a simple system of one degree of freedom, (2) the forced vibration 
of a stretched string at a point of which an obligatory motion is 
imposed, the damping in either case being assumed to be propor- 
tional to the square of the velocity. 

Case (i). The equation of motion of a system of one degree 
of freedom, subject to frictional forces proportional to the square 
of the velocity, and acted on by a periodic force sin pt % is 
given by 

0k v*+n*U=* sin pt (6) 

the alternative signs being necessary as otherwise the frictional 
forces do not change sign with the velocity. 

The solution of this equation will be of the form 

U=A } sin (#+,) +A 9 sin (jpl+*,) + A B sin (spt+*J + .... (7) 
Substituting (7) in (6) we get the resistance term as 

k &= kp*{A* cos* (/tf-H,) +(>A { A ?> cos (#+!) cos (3pt+*J 
+ 94 8 cos*(3# + s)+----} (8) 

Retaining terms proportional to A* and applying the Fourier 
analysis to get the periodic components of the frictional force, 
we get 

?* A* C 






(9) 



The series in (9) is a rapidly converging one. Substituting 
(9) in (6), expanding and equating the coefficients of sin pt and 
cos pt to zero, we get the following relations : 



The Forced Oscillations of Strings. 77 



O JL 

where A =n* - />* and , 

3*" 

The energy of the forced oscillation is a maximum when 
4 =o, i.e. when n=p. The amplitude is then given by 

These expressions show that the amplitude of the maintained 
oscillation at the peak of the resonance curve is proportional to 
the square root of the magnitude of the impressed force instead 
of the first power as in the ordinary case. Another special feature 
of the forced oscillation for this law of resistance is that the phase 
of the maintained motion, except at the peak of the resonance 
curve, varies with the magnitude of the impressed force, as can 
be seen from equation (ri). 

Defining sharpness of resonance as the square root of the 
quotient of the curvature at the peak of the resonance curve*, 
divided by the energy at the same position, we get 

Sharpness of resonance = - - 



So that we find that the sharpness of resonance is directly 
proportional to the frequency of oscillation of the system, and 
inversely proportional to the square root of the product of the 
impressed force and the damping coefficient, contrary to the case 
of ordinary resonance in which the sharpness of resonance defined 
in the same way is given by the equation f 

VL 

Sharpness of resonance = -- 

i 

We have seen that the resonance is maximum when np 9 



* The energy should be graphed as a function of the mistuniug (-^ -* \ 
\ E. H. Barton, Phil. Mag., July 1913. 



78 R N. GHOSH. 

but the amplitude of maintained motion is maximum when the 
frequency of the impressed force is equal to 

p { = (n* - A,a) (approx.) (13) 

and is therefore less than the natural undamped frequency of the 
system. The difference of the squares of the frequencies natural 
to the system and that for which the amplitude is a maximum, is 
approximately proportional to the product of the damping 
coefficient and the impressed force, whereas in the ordinary case, 
the difference of the squares of the above frequencies is propor- 
tional to the square of the damping coefficient. 
The maximum amplitude is given by 



* MI V) 

which shows that the maximum amplitude is proportional to the 

square root of the impressed force. 

When the amplitude of vibration is large and the restoring 

fore < cannot be taken as strictly proportional to the displacement, 

we may modify the equation of motion, and write 

Vku*+ (n*+PU*) C7=o sin pt (15) 

The displacement will be given by a series of the form 

17-4 / sin (pt+^ + AJ sin (3pl+* A ') + A b ' sin (5^+V) 
Applying the same method of analysis and proceeding as 

before we get equations similar to (10) and (il) etc., the only 

change necessary being made by writing A , for A in all the 

equations, where 



The effect of a finite amplitude is therefore equivalent to an 
increase in the natural frequency of oscillation of the system- 
Hence the maxima of all the resonance curves will not be found 
when n==p y but for large amplitudes, they will shift towards a 
lo\\cr natural frequency of vibration. 

Case (2). Now we shall pass on to the case of forced oscilla- 
tion of a stretched string whose every elementary portion is 
resisted by forces proportional to the square of the velocity. The 
equation of motion of the string is given by 



The Forced Oscillations of Strings. 79 

the alternative signs being necessary as explained before. The 
general solution of (16) is 

y=Wj sin pt-\-v { cos pt~}~u. A sin 3pt+v & cos 3/>-(-etc. (17) 
where the u's and v's are functions of x only. 

Substituting (17) in (16) we get the frictional force of the 
form 

kp* {u* cos 4 pt+v* sin* pt+ }, (approx.) 

Retaining terms proportional to u* and v* and analysing in 
Fourier series, we get 

kp* (u* cos* pt + v* sin' 2 pt} 
= -a p* (7** cos pt-\~v* sin />/+ terms of higher frequency) (iS) 

where = . 

Substituting (18) in (16) and equating the coefficients of sin pt 
and cos pt to zero, we get 



' dx* i , 

(19) 



The approximate solutions of (19) are 

w l = ^4j sin px/a - B*cL/2-\-Bf(*/6 cos 2 px/a-{- . . . 
t; l= Bj sin />AC/a - Afrfa + Afafi cos 2 px/a+ . . . 

from #=o to #=i. 

v y=i? sin 



. . B, sin px/a A fa/2 /, 

where tan f x = ' . ^ ', -L-i- = ^ approx. 

A } sin px'a Bfa/2 /., 

and Rt^fi+f*. 

From the conditions that at ^=&, y=y sin ^ the values of 
A { and jBj are determined. I^et us now take the important case 
in which the period of the forced vibration equals the natural 
period of the string ; then sin pb/a=o and all the expressions are 
much simplified. The expression for the displacement then 
comes out to be approximately 



8o R. N. GHOSH. 

y~ 
-^ sin pxja . sin 
a 

where tan <J> X = - I V sin pxla (20) 

Vya/ 

From (20) we see that the amplitude of vibration of the string is 
proportional to the square root of the obligatory displacement 
unlike the case when friction is proportional to the velocity, 
where the amplitude of vibration is directly proportional to the 
obligatory displacement. The motion of all points on the string 
except near the nodes is in approximately the same phase. The 
phase of the motion at the nodes is the same as that of the obli- 
gatory motion, while that of the rest of the string lags behind 
by quarter of an oscillation. The phase changes rapidly near the 
nodes, and two points of the string on opposite sides of the node 
at a sufficient distance from it generally move in opposite directions. 

The experimental results represented by fig. (4) agree with 
the theory according to which the maximum amplitude of vibration 
at a point for different values of y increases much less rapidly 
than in proportion to it. 

The effect of the finite amplitude of vibration can be easily 
calculated. The total length of the vibrating string at any in- 
stant is given by the formula 



/= 

J 

^l 









Whatever may be the nature of the function I ~), the effect of 

the finite amplitude is to increase the length. If, however, there 
be no longitudinal motion, it produces an increase in tension 
which would amount to a large value when the modulus of elasti- 
city is large. This increase in tension will therefore bring the 
points of maximum resonance towards lower tension and shift 
the whole resonance curve. This is the result actually obtained 



The Forced Oscillations of Strings. 8r 

as will be found from a reference to the experimental curve given 
in fig. (5). 

SECTION V. THEORY OF FORCED OSCIU,ATIONS UNDER 

L,ONGITUDINAI< EXCITATION. 

We shall now pass on to the case of forced vibration main- 
tained by forces of double frequency. The theory of oscillations 
under a periodic force varying the * spring * was first given by the 
late I/ord Rayleigh * on the assumption that the resisting force is 
proportional to the velocity. The theory has since been extended 
by Prof. C. V. Raman f who has shown how the equations may be 
modified and solved so as to give a determinate amplitude for the 
motion. 

CASE (I). Damping Proportional to the Velocity. 

This has been dealt with by Prof. Raman in the papers cited. 
The equation of motion of an oscillation of one degree of freedom 
under variable spring and subject to damping proportional to the 
velocity is 

~ sin 



The solution for the case in which the force has double the frequency 
of the maintained motion is 



U=A l sin (pt+tJ + A, sin ( 3 / + c g ) + 4 5 sin ( 5 ^+c g )+ etc. 
where A L and ^ are given as a first approximation, by 

(n 1 - p*+ & A *)*=** - k*p* (21) 

and tanv=(^)* (22) 



CASE (II). Damping Proportional to the Square of the Velocity. 
The equation of motion for this case is 

(Jku*+(n* - 2<* sin 2p*+/3[/*)i7=o. 
We assume the solution to be given by 

UA l sin (pt+ t ) + A sin ( 3 t+* + A sin 



* Scientific Papers, Vol. II, page 188. 

f Bulletin No. 6 of the Indian Association for the Cultivation of Science ; also 
Physical Review, Dec. 1912. 




82 R. N. GHOSH. 

Proceeding in the same way as before, we obtain the following 
relations as a first approximation 

(23) 
(24) 



It is important to notice the points of agreement and difference 
between the results in the two cases. 

The amplitude of vibration is not symmetrical with respect to 
*-/>* in both the cases. In case (i), that is with damping pro- 
portional to the velocity, we find that the amplitude of vibration 
increases continuously as (n* - />*) diminishes. This continues with 
(n* -p*) negative, and equation (21) does not put any limit to the 
continuous decrease of (* - p*) with consequent increase of the 
amplitude of vibration. It indicates that the tension can be 
diminished indefinitely, and the amplitude will then attain a 
correspondingly large value. But equation (23) of case (II) puts 
a limitation to the indefinite increase of the amplitude correspond- 
ing to an infinitely small tension. It shows that the amplitude 
of vibration is small when the tension is high, and increases with 
diminution of tension. The maximum value of the amplitude is 
not reached when (n ft - a )= Oj i.e. when the natural frequency 
of oscillation is half that of the maintaining force. When (n % - p*) 
becomes negative (the natural frequency of oscillation of the string 
is less than half that of the maintaining force) the amplitude 
increases still further. But the rate of increase of the amplitude 
in case (II) is not so rapid as in case (I). This is due to the fact 
that the right side of (23) goes on diminishing with the increase 
of the amplitude whereas the right side of (21) remains constant. 
This goes on till the right side of (23) becomes zero. At this 
stage the motion collapses, since the supply of energy becomes 
less than that lost by dissipation. The maximum value of the 
amplitude is, therefore, obtained by putting the right side of (23) 
equal to zero ; it is then given by 

A m "* a Sty* 



The Forced Oscillations of Strings. 



The same result may be obtained in another way. Solving equa- 
tion (23) we have 



where 
and 



2 &p + k. z and w=a* 
tt* - p* and 



The value of A when A f is maximum is given by 

dA^ . , cPB 

_- =o which gives d = - ~ 
d* & * 8 

The maximum value of A x is thus as before equal to 

A 3?ra 

Thus we find in case (II) there is a limit to the increase of the 
amplitude with diminution of n*-/> a whereas in case (I), equation 
(21) does not put any limit to the increase of the amplitude. 

The foregoing indications of theory agree closely with the 
observed facts. The continuous increase of the amplitude with 
diminution of tension has been found to be the same as required 
by the theory. The theoretical value of the amplitude when 
n=p is the same as found from experiment. 



t. 

r 



\ 



\ 



V 



FIG. 7. Resonance-Curve for Variable Spring of Double Frequency with 
Damping proportional to the Square of the Velocity. 



84 R. N. GHOSH. 

The theoretical graph fig. (7) plotted from equation (23) shows 
the relation between the amplitude of vibration of the string and 
n* - p*. The values of a and kp required for drawing the theoretical 
graph were determined directly from the known elasticity and 
damping of the string, and from the amplitude of the obligatory 
motion of the end of the string and the frequency of the motor 
vibrator. The agreement of the curves in figs. (6) and (7) is close. 
The amplitude is not symmetrical with respect to n 1 - p* as can 
be seen from the figures. The values of maximum amplitudes agree 
in the two curves. When n=*p, the amplitude is given by the 
point where the curve cuts the axis representing the amplitude. 
Both the curves agree here also. The rate of increase of the 
amplitude is not the same everywhere in fig. (7) : it is less near 
the collapse point than at small amplitudes of motion in ac- 
cordance with the observed fact indicated by fig. (6). 

Equation (21) of case (I) expresses the fact that in order to 
maintain the vibratory motion of the string, a must be greater 
than kp whatever the values of n and p, whereas equation 
(23) does not fix any minimum value for a. An attempt was 
made to maintain the motion of a long string with a very small 
magnitude of a, but it appeared that no maintenance was possible. 
This indicates that the frictional resistance for very small amplitudes 
is proportional to the first power of the velocity, and equations 
(21) and (22) then hold good; as soon as the amplitude of vibra- 
tion of the string becomes sensible, the observed facts are better 
in agreement with the theory worked out on the basis that the 
damping forces are proportional to the square of the velocity. 

The changes in the phase relation between the maintaining 
force and the maintained motion of the string may now be ex- 
plained. In case (I), equation (22) shows that when a, k and p 
are constant, i.e. when the amplitude of the motor- vibrator, its 
frequency, and the frictional coefficient are constant, the phase 
angle c should remain constant, whereas equations (23) and (24) 
of case (II) show that the phase angle will change even when the 
above quantities are constant if the permanent tension of the 
string and consequently its amplitude of vibration are altered. 
Equation (24) shows that as the tension is gradually decreased, at 
the initial stage where the amplitude of vibration is small, the 
phase angle varies very slowly, but when the amplitude is 
considerable, then it begins to decrease rapidly, and it continues 



The Forced Oscillations of Strings. 85 

to diminish, and attains the limiting value when ^i^cIT*' * e - 

when the motion of the string collapses. This is in agreement 
with observation. 

SECTION. SYNOPSIS. 

1. For large amplitudes of vibration it has been found that 
the frictional resistance to the vibration of a string in air is 
approximately proportional to the square of the velocity. 

2. The forced vibration of a simple vibrator resisted by 
forces proportional to the square of the velocity has been dealt 
with, and it has been found (i) the maximum resonance is pro- 
portional to the square root of the impressed force, (2) the sharp- 
ness of resonance is directly proportional to the frequency natural 
to the system, and inversely proportional to the square root of 
the product of the frictional coefficient and the impressed force. 

3. In the vibrations of a string obtaining by imposing a 
transverse obligatory motion at one point it is found that (i) the 
maximum amplitude of vibration increases much less rapidly than 
in proportion to the obligatory displacement. This has been 
theoretically accounted for by the square law of damping ; (b) 
the effect of finite amplitude of vibration in shifting the resonance 
curve towards lower tensions has been shown by graphs and 
explained. 

4. In the longitudinal type of Melde's experiment when the 
frequency of oscillation maintained is half that of the imposed 
variation of tension, it is found experimentally that (a) there is a 
definite limit corresponding to a particular magnitude of variation 
of tension imposed on the string, beyond which the amplitude of 
vibration cannot be increased by decreasing the constant part of 
the tension, the maintenance collapsing when this stage is reached 
and passed ; and (6), that the phase relation between the maintain- 
ing force and the maintained oscillation varies in different parts of 
the range of maintenance even when the imposed variation of 
tension is kept constant in frequency and magnitude. These 
facts cannot be fully explained if the frictional force resisting the 
motion of the string is assumed to be proportional to the first 
power of the velocity, but agree with the theory developed on the 
assumption that the friction is proportional to the square of the 
velocity. 



86 R. N. GHOSH. 

5. The experimental work described in the paper was carried 
out by the aid of the motor-vibrator designed by Prof. C. V. 
Raman, which was found specially suitable for the present investi- 
gation and indeed made it possible. 

6. The author hopes on a future occasion to develop and 
present the fuller theory for the types of maintenance with other 
frequency ratios. 



VL The Magneto-Crystalline Properties of the 
Indian Braunites. 



By K. Scshagiri Rao, B.A. (Hons.), Research Scholar 
in the Association. 



CONTENTS. 

SECTION I. Introduction. 

SECTION II. Physical Characters of Specimens Examined. 
SECTION III Experimental Methods and Results. 

SECTION IV. Chemical Composition of Braunite and Discussion of Results. 
SECTION V. Synopsis. 



SECTION I. INTRODUCTION. 

Braunite is a natural ore of manganese which according to 
Dr. Leigh Fermor, who has made an extensive study of the avail* 
able deposits,* is next to psilomelane the most important of the 
Indian manganese minerals and occurs in great abundance in 
various parts of the country. It is also found in other parts of 
the world both in crystallised and in massive form.f In spite of 
the enormous quantities of this mineral that are available it is 
only comparatively rarely that it is found in India in the form of 
measurable crystals. The detailed account of such occurrences 
will be found in the memoir already cited. A feature of this 
mineral which is not mentioned by Dana and to which attention 
has been drawn by Dr. Fermor is that it is invariably slightly 
magnetic.^ In view of the great importance of the mineral, it 
was thought that a quantitative study of its magnetic properties 
in the various occurrences in India would be of interest. 



* Memoirs of Geological Survey of India, Vol. XXXVII. 
f Dana's 'System of Mineralogy,' 1894 Edn., page 233. 
J Page 63 he. cit. 



88 K. SESHAGIRI RAO. 

SECTION II. PHYSICAL CHARACTER OF SPECIMENS EXAMINED. 

So far specimens from two localities have been examined by 
the author. 

The first specimen, which was very kindly furnished for the 
experiments by Dr. Fermor, was taken from the collection of the 
Geological Survey of India (No. K 348) and was from Kacharwahi 
in the Nagpur District in the Central Provinces. The specimen as 
received showed on the exterior a patch of white attached material 
which was a felspar and a small amount of decomposed blanfordite 
brown in colour. Except for these impurities, however, the whole 
specimen was part of one crystal showing well-cut faces of a deep 
steel-gray colour showing sub-metallic lustre, and when the 
specimen was ground down on a carborundum wheel to the shape 
of a sphere 2*4 centimetres in diameter, it was found that there 
was no visible felspar or blanfordite left on the surface. The 
sphere showed at eight regions on its surface (forming roughly 
the corners of a cube) a brilliant metallic glitter due evidently 
to the crystalline cleavages. It seemed thus quite certain that 
the sphere thus obtained was part of a homogeneous crystal. The 
material was hard and rather heavy, the density being 4*8. The 
Kacharwahi crystals belong to the tetragonal system. 

The other specimen was part of a hard lump of very pure ore 
picked up on the spot from the well-known hill at K&ndri, in the 
Central Provinces, which contains probably the finest body of 
manganese ore yet found in India and certainly one of the finest 
in the world. This is a fine-grained crystalline ore composed of 
apparently of braunite with a certain proportion of admixed 
psilomelane. A piece was broken out of this lump and also ground 
into a sphere of 2*4 centimetres diameter. It showed no visible 
impurity of an}' kind. The ore was very friable apparently owing 
to the admixed psilomelane. Its density was 4*22. 

SECTION III. EXPERIMENTAL METHODS AND RESULTS. 

The method of experimenting used was to grind the specimen 
into a sphere and determine the force pulling it into a non-uniform 
field, and from this force the average value of intensity of 
magnetisation for the sphere was calculated. If the susceptibility 
is constant throughout the spherical specimen, then to a first 
approximation the intensity of magnetisation calculated from the 



Magneto-Crystalline Properties of Indian Braunites. 89 

pull is the value of intensity at the centre of the sphere. This 
average value was the value that was determined. The advantage 
of the method is that one can easily test the homogeneity of the 
material. This may be done by simply inverting the sphere with 
reference to the field, in which case the pull will remain unchanged 
only if the lower and the upper half are alike. Within the limit of 
accuracy of this test the specimen was homogeneous. The expres- 
sion for the force exerted upon an inductively magnetised sphere 
along the y-axis is given by 



where F=Mechanical force, 

/Intensity of magnetisation, 
#=Magnetic field strength. 

If the magnetisation lies in the #y-plane the last term drops 
out and if the direction of the %-axis be taken as that of the 
field at the centre of the sphere, the second term will have a very 
small value, so that the final simplified result becomes 



To measure the force acting on the sphere, the following arrange- 
ment was adopted. A glass strip 2*5 cms. by 20 cms. was firmly 
clamped at one end to a rectangular brass piece fixed at right 
angles to a brass pillar moving up and down. The glass plate 
could be thus placed in proper position in the magnetic field. 
The other end of the plate carried a light wooden platform on 
which the sphere could be mounted. A pointer about 20 cms. 
long was attached to this end. In taking readings of the force 
the sphere was placed on the wooden platform. The bending of 
the glass plate as the field was slowly applied was measured by 
the motion of the end of the pointer which was observed through a 
microscope containing a scale in the eyepiece. The force acting 
on the sphere in dynes was calculated from the bending thus 
produced. For calibration the deflection produced by a known 
mass placed on the same position as the sphere was observed. 
With the arrangement used in the work a deflection of one division 
in the scale of the microscope eyepiece corresponded to about 
fifty dynes. 



go K. SESHAGIRI RAO. 

The magnet used in the work was a powerful electromagnet 
with large conical pole pieces which could be screwed into the 
iron cores of the magnet. These iron-cores moved in a groove in a 
strong iron bed plate and could be clamped in any desired position. 

The field strength was determined by an exploring coil con- 
sisting of 40 turns and an equivalent area of iO'2 square centimetres. 
The coil was connected to a ballistic galvanometer in series with 
a suitable resistance. The galvanometer throw when the field was 
suddenly introduced was noted. The galvanometer was standard- 
ised by a condenser of capacity O'2 microfarad. 

The field gradient was determined by determining the fields 
at two points which were a definite distance apart. 

Observations with the Specimen No. I. 

The sphere was first hung at random in a magnetic field by 
a silk fibre. A directive twist was observed showing that the sus~ 
ceptibility of the crystal varied in different directions. Now when 
a crystalline sphere having different susceptibilities in different 
directions is placed in a uniform magnetic field, a restoring couple 
due to this difference and clearly proportional to it tends to twist 
the sphere so that the direction of maximum susceptibility 
coincides with the direction of the lines of force. This direction 
of maximum susceptibility was determined in this way: 

First a uniform magnetic field was obtained by replacing the 
conical pole pieces by flat ones. Two circles at right angles 
to each other were marked on the sphere. The sphere was hung 
at random in this field and turned in its own plane along these 
circles at regular intervals. The frequency of oscillation was ob- 
served at each position. The positions on the two circles at which it 
was maximum were marked. It was evident then that these were 
the positions at which the restoring couples were maximum since 
the frequency of oscillation increases with the magnitude of the 
couple. Since the restoring couples are proportional to differences of 
susceptibility, the planes perpendicular to the axes of rotation con- 
taining these two points would each clearly contain the axes of 
maximum susceptibility. Hence the intersection of these would 
give the direction of maximum susceptibility. If this were made 
the axes of rotation the direction of minimum would be easily 
obtained. 



Magneto-Crystalline Properties of Indian Braunites. 91 

Tested in this way the sphere was found to possess a direction 
of minimum susceptibility. When it was mounted with this 
minimum as an axis of rotation the restoring couple became very 
small, showing the independence of magnetisation with direction 
in this zone. If the sphere were perfectly true it ought to rest 
easily in any position but as there were slight departures from 
complete sphericity and as the sphere could not be exactly placed 
in this position the couple did not absolutely vanish. 

The direction of the maximum may be somewhat more 
quickly obtained by hanging the sphere mounted at random in a 
magnetic field since the direction of the maximum and that of the 
field should coincide. By repeating this process the plane of maxi- 
mum susceptibility may be fixed. 

The sphere was then mounted on the plate and placed in the 
non-uniform field produced by the conical pole pieces. Readings for 
the force were observed for different fields. Now with a constant 

;\TT 

susceptibility the force exerted must vary as H -g- , i.e. as #*. 

Within the accuracy of measurements made it was found that 
both in the direction of the minimum axis and in the perpendicular 
plane that F/H* was constant, i.e. the substance was paramagnetic. 
The maximum field used was about 4000 gausses. 

Field and field gradient were measured by galvanometer deflec- 
tions. Field gradient was found nearly proportional to the field, 

j\TT 

i.e. H=K. y , and average value of K was used to calculate sus- 
ceptibility. The value of susceptibility was thus found to be 

0-4 x 10 ~ 3 . 

Readings for the force in the direction of axis of minimum 
and to the plane perpendicular to it were nearly identical. The 
difference of the susceptibilites was found to be too small to be 
exactly determined by the method adopted. An idea of the 
magnitude of this quantity was obtained as follows : 

The moment of the couple tending to turn the sphere in a 
magnetic about the axis of % from y towards z is given by 



Maxwell's Electricity and Magnetism, Vol. II. 



92 K. SESHAGIRI RAO. 

When referred to principal axes of magnetism the expression 
becomes 



YZ 



=*-', 

since when X<=0, Y=F cos 0, Z=F sin 0, provided is small. 

Thus by observing the time of oscillation when the sphere 
was mounted in a uniform field with its axis of rotation in the 
magnetic plane of symmetry and measuring the strength of the 

- Y <i 

field, ~^- 6t or the maximum difference of susceptibility was deter- 

mined in terms of the average susceptibility. This difference was 
found to be about 2% of the value of average susceptibility. 

The magnetic properties of the crystal as investigated thus 
bear a close resemblance to its physical characteristics. Braunite 
belongs to the tetragonal system, having a plane of symmetry 
perpendicular to the vertical axis. The value of f the vertical 
axis is given by Dr. Fermor as * 0*99. Magnetically the crystal 
has been found to possess a plane of symmetry at right angles to 
which is the axis of minimum susceptibility, the difference of 
susceptibility being of the same order as that of the difference of 
crystallographic axes. 

Specimen No. II. 

As the specimen No. II was not a part of a single crystal but 
only a homogeneous mixture of braunite and psilomelane, the 
determination of its susceptibility only was made. The observa- 
tions were made by comparison with those of specimen No. I. 
This was also found to be paramagnetic as in the case of No. I, but 
its susceptibility was only 75% of that of No. I. 

It would be interesting in view of the interesting magnetic 
property possessed by Indian braunites, to make a detailed 
investigation of European braunites which as far as one could 
gather from the literature have not received any attention. 



* Loc. cit. page 56. 



Magneto-Crystalline Properties of Indian Braunites. 93 



SECTION IV. CHEMICAL COMPOSITION. 

Both of these specimens were analysed in the chemical labora- 
tory of the Association. The results of these analyses are shown 
below : 





Kacharwahi. 





Iron 

Manganese 

Silica and insoluble residue 



Kandri. 



54% 



2*6 



12-6 



iron. 



Both the specimens contain a relatively low proportion of iron. 
Analysis of these specimens as recorded by Dr. Permor 
also given below for comparison. 



are 





Kacharwahi. 


Kandri. 


Manganese 


57-86 


58-55 


Iron 


3-85 


3*37 


Density 


4'79 


4-28 



It seems not unlikely that the magnetic property of braunitc 
may be largely due to its manganese content, though on account 
of the presence of iron oxides in the substance it is difficult to 
pronounce decidedly on this point at present. 

SECTION V. SYNOPSIS. 

The results of investigations upon the magnetic properties of 
Indian Braunites may be stated as follow : 

(1) Symmetry of braunite is magnetically that of an axis of 
symmetry, with a plane perpendicular thereto in which the mag- 
netic susceptibility is independent of orientation. 

(2) Along this axis of symmetry the susceptibility is minimum 
and in the plane it is maximum, the difference between the two 
being about 2%. 

(3) Both in this plane and along the axis it is paramagnetic, 
the susceptibility being very low. 



94 K. SESHAGIRI RAO : Indian Braunites. 

(4) All these magnetic properties bear a close resemblance to 
the physical characteristics of the crystal which are a crystallo- 
graphic plane of symmetry at right angles to the vertical axis, the 
value of the axis <! being 0*99. 

In conclusion, the writer wishes to thank Professor Raman 
who suggested the investigation and took much interest in its 
progress. 



VIL Free and Forced Convection from Heated 
Cylinders in Air* 



By Bldhubhusan Ray, M.Sc., Palit Research Scholar in the 
University of Calcutta. 



(Plate VI). 



CONTENTS. 

SECTION I. Review of previous work on convection. 

SECTION II. Optical study of convection by the Foucalt method. 

SECTION III. Electrical determination of temperature distribution in free 

convection. 
SECTION IV. Synopsis and discussion of results. 

SECTION I. REVIEW OF PREVIOUS WORK ON CONVECTION. 

The phenomenon of convection from the surface of a heated 
body immersed in a fluid has received considerable attention both 
from the theoretical and the experimental points of view. If the 
cooling fluid be a gas, then, the variations of pressure, density, 
velocity, thermal conductivity and viscosity at different points, 
so complicate the problem that not only has little progress towards 
a complete mathematical solution yet been made, but for some 
time a true knowledge of its laws seemed nearly impossible. 
A. Oberbeck * gave the general differential equation for this problem, 
but being unable to solve it generally was satisfied with working 
out a solution for a special case. Later on, L. Lorentz f obtained 
a solution for the case of a vertically placed plane strip cooling in 
air. When the strip is protected from draughts, he found that 



* Ann. Phy. 7, 271 (1879). f Ann. Phy. 13, 582, 1881. 



96 B. RAY. 

the heat converted away from the surface varies as ' 9 where 6 is 
the difference of temperature between the strip and air before it is 
heated by the strip, and this value agreed fairly well with some 
of the older experimental work. H. Wilson* attempted to solve 
the problem taking the viscosity into account but his solution is 
limited to a very special case. 

In 1901 Boussinesq f took up the problem and first clearly stated 
the laws for the forced convection of heat for a body immersed in a 
stream of liquid when the flow is not turbulent. The same author 
after four years published a memoir containing a great number of 
successful solutions of the problem of the convection of heat from 
bodies of various shapes immersed in a stream of fluid. It is to be 
noticed here that according to Boussinesq 's result, the heat loss due 
to convection of heat from bodies immersed in a stream of fluid varies 

as V*0 y where V is the velocity of the current and is the differ- 
ence of temperature of the solid body above that of the surround- 
ing fluid at a great distance. Eennelly.f making elaborate 
measurements of the convection of heat from thin copper wires, 
remarked as follows: "The lateral conduction through air is 
negligible because the air does not remain at rest but expands 
and flows convectionally. Consequently we may safely ignore 
conductive thermal loss. Convection loss from the air is a 
hydrodynamical phenomenon, involving the flow of air past the 
surface of the wire and the amount of heat which the moving 
stream can carry off. Very little seems to be known quantitatively 
about convection.' 1 Being unable to find any general solution, 
he was therefore satisfied with framing an empirical law to express 
his results. In the case of forced ccnvection of heat his results 
agreed with those of Boussinesq for the cooling of cylindrical 
wires. 

Russel, following after Boussinesq, discussed the problem at 
some length. In order to solve the general differential equation 
and apply its results to bodies of various shapes and dimensions, 
Russell was compelled to make certain assumptions with regard 
to the nature of the fluid sorrounding the solid. The liquid is 



* Camb. Phil. Soc. Progs. 12, 406, 1902. 

t Comptes. Rendus. Vol. 133, p. 257. 

% Trans. Amer. Inst. E.E. 28, 263, 1909. Phil. Mag. 20, 591, 1910. 



BIDHUBHUSAN RAY. 



PLATE VI. 




FIR. I 

Free Convection 




Fig. 9. 

Forced Convection 



CF 




Fig. 3 

Forced Convection 



Free and Forced Convection from Heated Cylinders. 97 

supposed to be opaque to heat rays, it has no viscosity and so it 
slips past the surface of the solid, also it is supposed to be incom- 
pressible. The further assumptions are, " The thermal conductivity 
of the liquid is very small and the variations in its density does not 
appreciably alter the trajectories of the liquid particles in the imme- 
diate neighbourhood of the solid from the shape they have during 
isothermal flow. The former assumption is true in most practical 
cases, and the latter is permissible when the velocity of the current 
is appreciable and no eddies are formed. The surface of the solid 
being cooled by the current is supposed to be isothermal, and the 
liquid in immediate contact with it at any instant is supposed to have 
the same temperature as the solid." In view of these very impor- 
tant assumptions we should expect the solutions to give only approxi- 
mate values when applied to the problem of spheres and cylinders 
cooled by currents of air. The result, which Russell arrived at, 
and which is true for the two-dimensional flow round a solid of 
any shape immersed in a stream of liquid, agrees with that of 
Boussinesq ; and when particular cases are considered it agrees 
with the work of Kennclly,* Ayrton and Kilgour | and other 
workers. All the investigators in calculating the heat loss from 
the surface of the body find it is proportional to the difference of 
tern pcrature between the solid and liquid surrounding it. It is to 
be noted that Newton's law of cooling is thus verified when the 
cooling fluid is a liquid, and Newton himself enunciated his law 
with reference to the convection and not the radiation of heat. 
In several practical applications, Newton's law of cooling 
leads to results agreeing closely with experiment. In the theory 
of a hot-wire oscillograph Irwin J has assumed that the convection 
loss of heat from a heated strip immersed in oil is proportional 
to the difference of temperature between the metal and the oil. 
The very satisfactory coincidence between the theoretical and 
experimental result shows that the assumption is approximately 
correct. 

Russel in the same paper shows that in the case of turbulent 
motion of water through a pipe, Newton's law is very approxi- 
mately true. It is to be noted here, that we are now discussing 



* Journ. Inst. Elect. Engin. 35, 364, 1905. 
t Phil. Trans. Vol. 183, Pt. i, p. 371, 1892. 
J Journ. Inst. Elect. Engin. 39, 617, 1907. 



9 8 B. RAY. 

the problem of forced convection of heat, but in the case of free 
convection Newton's law of cooling is not applicable in liquids, 

where Lorentz's law that the heat loss in proportional to *, 
where is the difference of temperature between the strip and air 
before it is heated by the strip is applicable. 

I. Langrnuit * differed from the previous workers in two of 
the important assumptions, namely in the effect of viscosity and 
the variation of the thermal conductivity due to temperature. 
This author contends that in the case of convection from small 
wires the effect of viscosity is most important. His own views are 
clearly given thus in his thesis on some reactions around glowing 
Nernst filaments: " According to the kinetic theory, the viscosity 
of gas increases with square root of the absolute temperature, 
the driving force of convection being proportional to the difference 
in density between the hot and cold gas increases only very 
slowly with increasing temperature. Therefore in the immediate 
neighbourhood of the filament the flow of gas is small and the heat 
must be carried away practically only by conduction. It is highly 
probable that at a very high temperature, the motion of a gas m 
the immediate neighbourhood of the wire would not perceptibly 
increase but probably decrease, while at the same time the heat 
conductivity of the gas would increase very greatly. Thus at a 
distance very near the surface of a body, heat conduction is more 
important than convection ; so that heat will be carried only by 
radiation and conduction but not by convection." Several con- 
siderations had made it seem probable that the above theory would 
be fairly close to the truth, For instance it was found that the 
watts loss from a wire was the same, whether the wire was placed 
horizontally or vertically. Now the Hues of flow of the heated air 
around the wire would be totally dissimilar in these two cases. 
Yet it was found that the energy necessary to keep a piece of pure 
platinum at a given temperature the resistance being kept 
constant never differed appreciably for horizontal or vertical wires, 
and at a bright red heat or above, the difference became negligibly 
small. This was a strong indication that the heat loss was depen- 
dent practically on heat conduction very close to the filament and 
that the convection current had practically no effect except to 

* Phys. Rev. 34, p. 401, 1912- 



Free and Forced Convection from Heated Cylinders. 99 

carry off heat away after it has passed out through the film of 
adhering gas. The thickness of the film of gas through which 
conduction takes place can be calculated if the temperature of the 
wire, its diameter, and the heat conductivity of the gas are known. 
The last factor varies very much with temperature. 

lyangmuir, in calculating the heat loss from a heated wire, 
assumes that a thin cylindrical film of gas adheres to the wire 
through which heat travels only by conduction and radiation, and 
the thickness of the film of air is the distance heat must travel 
before the heat flux due to temperature difference becomes small 
compared with that due to convection. Throughout this film of air 
the viscosity, the thermal conductivity and specific heat of air 
varies greatly. Meyer * has given a formula for conductivity in 
terms of viscosity and specific heat (per gram) at constant volume, 
while Sutherland obtained an equation connecting viscosity and 
temperature containing two constants, which have been determined 
by Raiikine.f M. Pier i has obtained an equation for specific heat at 
constant volume in terms of temperature, and so connecting these 
formulae, T v angmuir obtained a relation between conductivity and 
temperature, and applying the simple conduction formula in the 
thin film, he calculated the heat loss from the heated wire. His 
experimental result agrees fairly well with the theoretical formula. 

Later on, the same author published an account which is an 
extension of his earlier work to cover convection of heat from 
plane surfaces. Experiments are described in which the heat loss 
from plane surfaces at various temperatures is determined. After 
discussing the results of this experiment with his previous theory, 
he finds that the film theory is found not to apply to convection 
forced by air currents, but Russell's formula agrees well with the 
experimental data. Morris || has verified the application of the 
formula of the type obtained by Boussinesq to the cooling of fine 
wires heated by electric current to temperatures about 7Oc above 
the surrounding air and for air velocities as high as 40 miles per 
hour. 

Prof. L. V. KingU follows Boussinesq and transforms the 



* Kinetic Theory of Gases. t Proc. Roy. Soc. Ion. A84, 181, 1910. 

Z. f. Electrochem. 15, 536, 1909 and 16, 899, 1910. 

Amer. Electrochem. Soc. Trans. 23, p. 299, 1913. 

|| Electrician, Oct. 4, 1912 ; also Engineer, September 27, 1912. 

If Phil. Trans. A, Vol. 214, 1914. 



TOO B. RAY. 

general equation of conduction into a greatly simplified form in 
the case of a liquid flowing past a cylinder, reducing the problem 
to the case of a two-dimensional flow. The problem can now be 
transformed into a form having a known partial differential equa- 
tion. The complete solution of this equation requires a know- 
ledge of the conditions of heat transfer over the interface between 
the solid and the liquid. "The solution of this problem which 
gives results in best agreement with experiment for the case of 
convection of heat from small cylinders is that obtained by assum- 
ing the flux of heat to be constant over the boundary. As a 
result of high heat conductivity of the cylinder in the experiment 
carried out, the temperature of the cylinder may be considered 
constant over the boundary and there will therefore be a discon- 
tinuity in the temperature of the boundary ; we assume that the 
temperature of the stream in contact with the cylinder becomes 
finally equal to that of the cylinder at the point where it leaves 
the boundary/ 1 The result he obtained for the loss of heat for 
small and large velocities of the air current agrees fairly well with 
experiment and with that of some of the previous workers in the 
case of forced convection. It is to be noted here that King based 
his theory on simple hydrodynamical flow in which there is 
slipping at the boundary between solid and liquid. Here he 
takes no account of viscosity, variation of thermal conductivity 
and specific heat near the surface. It is interesting to note that 
he gets the value of some constants which agrees fairly well with 
that of the other investigators working on different lines. 

Kennelly * later on found, that- for pressures between 5 and 
3 atmos., the rate at which heat is dissipated varies not only as 
the square root of the speed of the wire as already found out but 
also approximately as the square root of the air pressure in accord- 
ance with the theoretical result of Boussinesq. 

V. M. Schidlovski f in testing the truth of Iy. Lorentz's J for- 
mula for the case of cooling of a wire finds that the current 
sufficient to heat the vertical wire to redness leaves the horizontal 
one dull, and the effect due to conduction and convection is less in 
the case of vertical wire than when it was horizontal, and this 



* Amer. Phil. Soc. Proc., Vol. 53, 55, 1914. 
f Russian Physico-cheni. Soc. J., 43, 132, 1914. 
J Ann. Phys. 13, 582, 1881. 



Free and Forced Convection from Heated Cylinders. 101 

decrease is due to the ascending hot-air currents enveloping the 
wire. 

T Barret *, while experimenting on radiation and convection 
from a heated wire in an enclosure of air, finds that as the tem- 
perature is raised the heat loss due to convection increases more 
rapidly than that due to radiation. 

I. A. Hughes f has measured the loss of heat per c.m. length 
and also the loss at different velocities in the case of cooling of 
cylinders in a current of air, his results agreeing with those of the 
previous workers. 

W. P. White J has made an investigation to obtain data for the 
use in the design of calorimeters. According to him, leakage due to 
conduction and radiation conforms substantially to Newton's law of 
cooling. Heat transfer by convection, being due to air currents, 
whose temperature and velocity are both affected by thermal head, is 
more nearly proportional to the square of the temperature differ- 
ence. The observed variation from Newton's law of cooling is to 
be ascribed to convection. He appears to have worked with 
cooling of bodies with small air gaps, so his result is not compar- 
able with other workers who have worked with an infinite air gap. 
The results obtained both for eddyless motion and for turbulent 
flow are intended to serve as a practical guide in calorimeter 
designing. 

In view of the fact that different (sometimes opposite) 
assumptions have been made by different investigators regarding 
the nature of the boundary conditions at the surface of the solid, 
and that there is much regarding the subject which is still obscure, 
the problem was taken up to study more fully 

(1) the exact nature of the convection phenomena, and 

(2) the temperature and velocity distribution round the 
heated body, and to find the law of flow both for free and forced 
convection of heat and to obtain material for dealing with the 
problem on a stricter hydrodynamical basis. 

SECTION II. STUDY OF CONVECTION BY THE FOUCAI/T METHOD. 

Some idea as to the nature of the convection phenomena 
may be formed by using smoke or some such substance round the 

* Lond. Phys. Soc. Proc. 28, p. i, 1915. 

t Phil. Mag. 33, 118, 1916. J Phys. Rev. 10, 743, 



102 B. RAY. 

heated body, and allowing strong light to fall on this smoke which 
becomes luminous and makes the nature of its movement 
visible to the eye. A much better method which has been used 
by the present author is an optical method of observation which 
is a modified form of the Foucalt test or Topler Schliereri appara- 
tus. According to the simple laws of geometrical optics, rays 
issuing from a point can be focussed at another point if the 
optical instruments are perfect. An eye situated just behind the 
focus observes an illuminated field, but if a sharp-edged screen be 
gradually advanced in the focal plane, all light is suddenly cut off 
and the entire field becomes dark simultaneously. At this mo- 
ment if there be any irregularity in the optical surfaces or in the 
medium through which the rays come to a focus, the rays arc 
deviated from their proper course so as to escape the screening 
and the field becomes luminous. From the picture of the lumin- 
ous patch the nature of the irregularity of the optical appliances 
or of the medium through which the rays are passing can be is 
studied. 

The arrangement of the experiment is as follows : A copper 
cylinder of radius 7*2 millimeters is used for the experiment. A 
heating coil made of nickel-chromium wire, wound on a frame 
of mica, similar to that used in platinum-resistance thermometers, 
is inserted into the cylinder, which is kept in a horizontal 
position b} means of two stout wires as seen in the photograph. 
A current of i'8 amp. is passed through the coil, and the 
cylinder which was insulated is thereby heated. 

White light from a small circular hole, illuminated by the 
electric arc, is incident upon a good achromatic lens, which forms 
a sharp image of the hole at a considerable distance from the lens. 
A thin circular metal disc fixed on a thin sheet of mica is put in 
the focal plane, and the whole arrangement is so adjusted that 
the metal disk just completely cuts off the image of the circular 
hole formed by the achromatic lens. A telescope is placed with its 
object-glass just behind the metal disk attached to the sheet of 
mica and is pointed towards the lens. The copper cylinder from 
which the convection is observed is placed immediately in front 
of the lens with its axis along the optical axis of the system, and 
the observing telescope is focussed on the end of the cylinder. 

When the cylinder is heated by passing the electric current, 



Free and Forced Convection from Heated Cylinders. 103 

the field of view of the telescope which is previously just moder- 
ately dark soon begins to show the luminous streams of air 
rising from the cylinder. The phenomenon seen gives a vivid 
visual picture of the nature of the temperature distribution round 
the cylinder and of the motion of the air. 

Figure i in Plate VI illustrates free convection from the 
heated cylinder. The luminous patch seen is very broad and 
diffuse, and quite steady. The horns pointing upwards show the 
path along which the heated air rises. It will be seen that some 
distance above the cylinder and between the two horns we have a 
distinct darker vertical strip in which the temperature appears to 
be less than within the horns. 

Figure 2 in Plate VI shows the effect observed when a gentle 
current of air from an electric fan is forced horizontally at right- 
angles to the cylinder. The horns of the luminous area are here 
much better defined, and it was noticed that they were somewhat 
unsteady in position owing to the fluctuations in the current of 
air. As visually seen the thin tips were much more pointed than 
would appear from the photograph. The photograph was taken 
with quite a short exposure. The luminous patch here is much 
sharper and was visually observed to be far more brilliant than in 
the case of free convection, showing that the gradient of tempera- 
ture is greatly increased by the forced draught. 

Figure 3 in Plate VI also shows a case of forced convection, 
the velocity of the draught being greater here than in the case of 
Fig. 2. The features mentioned above, are here seen in more 
accentuated form. A considerable difference in the thickness, 
brightness and sharpness of the luminous fringe in front of and 
behind the cylinder will be noticed. 

In the photograph reproduced in the Plate, one of the 
luminous horns is seen to be distinctly brighter than the other. 
This is due to the fact that the circular metal disk which cut off 
the image of the source was not exactly symmetrically placed. It 
was very interesting to watch the luminous streamers on the 
ground glass of the camera when the position of the disk was 
slightly altered or when irregular draughts of air were allowed to 
fall on the cylinder. The adjustment of the disk to the final 
position of symmetry had to be made by actual trial. In studying 
the free convection, it was found necessary to protect the cylinder 



104 



B. RAY. 



from irregular draughts by surrounding it by a large wooden 
box. 

Observations were then made of the diameter of the luminous 
fringe surrounding the cylinder and of the velocity of the wind. 
It was found that as the velocity of the wind was increased the 
diameter of the patch decreases at first very rapidly but after a 
time very slowly until finally with the increase of the velocity 
of air, it is difficult to detect any change in the width of the lumin- 
ous patch. The following measurements of the diameter of the 
patch give an idea of the effects observed : 





Velocity of the wind. 


Angle width of patch in the field of the 
observing telescope. 


Free 


convection 


O-i4 / -44" 


5-6 ft 


. per min. 


Q f > 12' 30" 


8-5 ft 


per min. 


OP-H'-aT 


9'8 ft 


. per min. 


0-0'-43- 


12-5 ft 


. per min. 


00-8'- 47 


18-2 ft 


. per min. 


0-8'-iO* 



SECTION III. ELECTRICAL DETERMINATION OF TEMPERATURE 
DISTRIBUTION ROUND THE CYUNDER IN FREE CONVECTION. 

To confirm the general indications furnished by optical observa- 
tion, the distribution of temperature in the region surrounding 
the cylinder was studied by an electrical method. A thermocouple 
of fine copper and constantan wires was used, the junction 
between them being made by electrical fusion. Care was taken to 
see that the junction of the two wires occupied a very small region. 
One end of the couple was placed in ice and the other placed near 
the cylinder. The position of this end with respect to the distance 
and direction from the cylinder could be determined by means of a 
telescope having a millimeter scale in the eyepiece. The thermo- 
couple was balanced against a calibrated potentiometer bridge wire 



Free and Forced Convection from Heated Cylinders. 105 

and the balance point carefully noted. The cylinder was protected 
by a wooden box enclosing it, and very great care had to be 
exercised to avoid disturbing currents of air. 

Five sets of readings for each position of the couple were 
taken and the mean of the five was used for drawing the form of 
the isothermals round the cylinder. 



rnm.zo 




18 /> 6 6 /2 18 milli metti* 

FIG. 4. Temperature Distribution around Heated Cylinder. 

The curve (Fig. 4) shows a striking agreement between the 
temperature distribution round the heated cylinder as observed 
electrically and as indicated by optical observation and illustrated 
in Fig. i in Plate vi. 

It is to be noticed that the junction of copper-constantan 
occupies a finite space and hence when it is placed near the 
cylinder, the flow of heat round the cylinder is disturbed by its 
very presence. Some heat will be carried along the wires away 



106 B. RAY. 

from the junction and some heat will be radiated away from the 
cylinder to the junction. These effects cannot be entirely 
eliminated but may be diminished by making the junction occupy 
a very small space and having the wires of the couple made very 
thin. 

One important point to be noticed is that the isothermals are 
crowded together on the lower side of the cylinder, and diverge on 
the upper side of the cylinder. 

vSECTiON IV. SYNOPSIS AND DISCUSSION OF RESULTS. 

The present paper describes the results of an experimental 
study of the free and forced convection of heat from horizontally 
held cylinders in air. The general character of the temperature 
distribution round the cylinder was optically observed by the 
method of the Foucault test and quantitatively determined by 
thermo-electric measurement of the temperature of the air. The 
following are the principal features observed : 

1. In free convection, there is a relatively large region of 
heated air surrounding the cylinder, the temperature gradients are 
relatively small, and the region above the middle of the cylinder 
is actually cooler than the region vertically above its two edges. 
Immediately on the surface of the cylinder itself, the gas appears 
to be quiescent, the movement increasing as we pass away from 
the cylinder on either side. There is probably no actual discon- 
tinuity of temperature at the boundary itself, but there is a rapid 
fall of temperature as we move away from the surface. 

2. In forced convection, the layer of heated air is much 
thinner, and the temperature gradients are accordingly greater. 
There is a noticeable difference in the thickness of the heated 
fringe of heated air on the leeward and windward sides of the 
cylinder, and this thickness decreases with increasing velocity of 
the air current. The ' horns ' or currents of heated air moving 
away from the cylinder are much sharper in forced convection 
than in free convection. 

3. The optical observations and the experimentally deter- 
mined form of the isothermals in the fluid surrounding the cylinder 
suggest the question as to how far the hypothesis of King that 
there is a discontinuity of temperature between solid and the 
surrounding gas represents actual fact. The evidence suggests 



Free and Forced Convection from Heated Cylinders. 107 

rather that in the case studied by the present author that there is 
little or no slipping between the fluid and the solid at the 
boundary, and no actual discontinuity of temperature though there 
is a rapid fall of temperature close to the boundary. Though 
there is no actually stationary film of finite thickness of gas in 
contact with the solid , there appears to be very little motion in the 
immediate neighbourhood of the boundary, and presumably at the 
boundary itself the gas is at rest. By considering the effect 
of viscosity of the fluid, it would seem possible to explain the 
effects observed. The mathematical theory of convection contained 
in King's paper appears to require modification taking the effect of 
viscosity into account to enable the observed form of the isother- 
mals to be explained, and the agreement obtained between King's 
theory and the observed convection-losses of heat from small 
cylinders may require to be interpreted afresh. 

The experimental work of finding the distribution of tem- 
perature round the cylinder was carried out in the Palit Laboratory 
of Physics, and the author offers his best thanks to Prof. C. V. 
Raman, Palit Professsor of Physics, for suggesting the problem and 
for his lively interest in the progress of the investigation. 

University College of Science, 
Department of Physics, 

92, Upper Circular Rd., Calcutta, 
1920. 



VIIL Experiments on Impact. 

By A. Venkatasubbaraman, B.A. 



In a paper which has appeared in the Physical Review for 
April 1920, Prof. C. V. Raman has shown how the well-known theory 
of impact developed mathematically by Hertz may be extended and 
applied to the problem of the transverse impact of a sphere on a plane 
elastic plate of finite thickness. The theory involves the considera- 
tion of the proportion of the kinetic energy of the impinging 
body transformed into energy of wave- motion in the elastic plate. 
Starting with the assumption that, to a first approximation, the 
duration of impact is the same as on Hertz's theory for the case 
of an elastic plate of infinite thickness, the potential and kinetic 
energies carried off by the flexural waves generated in the plate 
by impact are calculated 011 certain simplifying assumptions 
regarding the form of these waves. The result finally obtained is 
that, e } the coefficient of restitution of the impinging body is 
given by the formula 



//>a* + o-56 M. 
where 



T is the duration of impact, 
E is Young's Modulus for the plate, 
o is Poisson's ratio for the plate, 
2/ is the thickness of the plate, 
p is the density of the material of the plate, 

and M is the mass of impinging body. If T (the duration of 
impact) be taken to have the value given by Hertz's theory for a 
plate of infinite thickness, we may expect a good agreement be- 
tween the formula and the observed coefficient of restitution for 



no A. VKNKATASUBBARAMAN. 

moderately thick plates. For thin plates, it would be more nearly 
correct to take T as the actual duration of impact which would no 
doubt differ somewhat from that for the case of an infinitely thick 
plate. 

To test the foregoing formula, the author has made a series 
of observations using a set of polished hard steel balls of various 
diameters and a set of glass plates of different thicknesses. These 
materials were chosen as, provided the size of the impinging sphere 
is not too great and the velocity of impact is not too large, no 
permanent deformation of the impinging bodies results from the 
impact. For velocities of impact greater than a certain limit 
depending on the size of the impinging sphere, the collision results 
in the formation of percussion-figures of beautiful geometrical 
form which have been observed and described by Professor C. V. 
Raman (' Nature,' Oct. 1919). If a percussion-figure be formed, 
we cannot naturally expect the formula for the coefficient of 
restitution to remain valid, and it is accordingly necessary to 
restrict the comparison with it to cases in which the spheres and the 
velocities of impact are moderately small. 

The accompanying table gives the results for the coefficient 
of restitution obtained by the writer for a certain moderate velo- 
city of impact. The coefficient of restitution calculated from the 
formula are also shown, T being taken in the formula to be the 
duration of impact as given by Hertz's theory for an infinitely 
thick plate. 

The following facts emerge on an examination of the figures 
shown in the Table of results. For the thicker plates, the experi- 
mental values for e are smaller by two or three per cent, than 
the theoretical values. This is evidently due to various minor 
causes of dissipation of energy not being taken into account in 
the theoretical treatment. For moderately thick plates, the 
calculated and observed coefficients of restitution agree well. 
Theory and experiment also agree in the case of very thin plates 
in giving a zero coefficient of restitution. In other words, in such 
cases, the sphere on impact with the plate remains in contact with it 
and behaves nearly in the same way as a perfectly inelastic body. 
But in certain intermediate cases where the thickness of the plate 
is less than about half the diameter of the impinging sphere but 
not so small as to give a zero coefficient of restitution, the observed 



Experiments on Impact. 



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ii2 A. VENKATASUBBARAMAN. 

values of e are somewhat larger than the calculated values. This 
appears to indicate that in the case of such plates, the actual 
duration of impact is somewhat greater than that given by Hertz's 
theory for the case of an infinitely thick plate, and that better 
agreement between theory and experiment would be obtained if 
such actual duration was taken as the value of T in the formula. 
(Of course, this would only be permissible provided the actual 
duration is finite and not very much greater than given by Hertz's 
formula.) 

Some observations made by the writer show that when the 
velocity of impact on the plate is so great as to result in the for- 
mation of percussion-figures, the coefficient of restitution falls 
considerably below the value given by the formula above. This 
is what may be expected in view of the fact that the production of 
the percussion figures must result in dissipation of energy which 
involves a decrease of the coefficient of restitution. In this connec- 
tion, it is also worthy of note that much higher falls and therefore 
larger velocities of impact may be used in the case of small spheres 
than in the case of larger spheres without resulting in production of 
percussion-figures. As the magnitude of the stresses per unit area 
set up by impact is (according to Hertz's theory) the same for the 
large as well as foi small spheres, this observation suggests that the 
duration of impact is distinctly a factor in determining whether or 
not the impact should result in the production of a percussion figure. 
The author has also observed that the area of the percussion-figure 
increases appreciably with increasing velocities of impact, and also, 
as might be expected, with increasing size of the impinging sphere. 
From the observations of the decrease of the coefficient of resti- 
tution, it is possible to calculate the energy required for the produc- 
tion of the percussion-figure and to find how it varies with the 
area of the internal fracture which constitutes the percussion- 
figure. 

The author has also found that appreciable deformations from 
planeness of the surface of the glass plate result from the produc- 
tion of percussion-figures and may be detected by optical methods. 
This phenomenon is no doubt intimately connected with the 
question of residual strains in the plate left after production of 
the internal fracture. 



IX. The Theory of the Flute. 



By Dr. G. T. Walker, C.S.I., Sc.D., F R.S., Director-General 
of Observatories In India. 



Among the more attractive applications of mathematical 
physics are those which have been made to musical instruments; 
and although in most cases the main principles have been truly 
laid down there are details still unexplained which may be of some 
theoretical or practical interest. Some recent experience in 
designing piccolos of different bores has brought out various 
points in which the existing theory of the flute appears incomplete, 
and I propose to give a short account of these in the hope that 
it may stimulate a further theoretical examination of the subject. 
An excellent account of the many papers dealing with the 
acoustics of pipes will be found in Winkelmann's Handbuch der 
Phy-ik (Vol. II, pp. 446, 7. Leipzig, 1909). 

2. The questions at issue fall naturally into four classes as 
they are concerned with (a) the shape of the bore of the instru- 
ment; (b) tlie position of the holes and open end; (c) the size of 
the holes ; (d) the method of blowing ; and (e) the method of finger- 
ing. Considering first the bore, Boehm pointed out that while 
a flute with a large internal diameter possessed facility and tone 
in the lowest notes the highest notes were unmanageable ; and 
of an inch or 19 mm. is generally accepted as the standard for the 
ordinary D flute whose length is about 24 inches. For the 
ordinary D piccolo a bore of " gives a rather feeble lowest octave 
and the usual bore of T V' is better : one of \" is brilliant in its 
lowest notes, but rather difficult in the highest. Boehm also 
pointed out that a flute with a purely cylindrical bore cannot be in 
tune throughout its range, the notes of the second octave being 
flat by comparison with those of the first and those of the third 
by comparison with those of the second. It would thus 



ii4 O. T. WAT.KTCU. 

appear that the correction for the open end or for a large hole 
in a cylindrical tube increases appreciably with the frequency of 
the note. 1 Boehm rightly gave as tho remedy for this difficulty 
a constriction of the bore of the upper portion of the tube (that 
containing the mouth-hole) so that the bore diminishes gradually 
from its full dimensions at a quartei of the length from the top 
end to nine-tenths of the dimensions at the mouth-hole. Th r> 
same result is apparently produced in a conical flute by making 
the top third cylindrical and the lower two-thirds conical, the 
diameter at the open end being about "7 of that of the cylindrical 
portion. 

3. The length of the tube to produce a given note will differ 
from the half of the corresponding wave length by the sum of the 
corrections for the mouth-end of the flute and for the open end. 
Similarly the positions of any of the holes will depend on the sum 
of the corrections for the corresponding hole and for the mouth- 
end. An obvious way of determining the facts is to measure the 
distances of a series of holes of equal size from the mouth-hole in 
an actual flute. When the unknown but constant correction has 
been subtracted these distances must bear to the unknown distance 
for the fundamental note a series of known ratios depending on 
the musical intervals of the notes in question. We thus have a 
number of equations with only two unknowns, these being the 
constant correction (the sum of the corrections for the mouth end 
and for a hole of the given sue) and the distance for the funda- 
mental note. In this way I have deduced from my flute, with a 
bore of ", that the distances of the |" holes from the mouth-hole 
are less by 2*40" than their position if there were no corrections ; 
for a f" hole the corresponding amount is 2*27". Also the length 
from the mouth-hole to the open end is less by 2'o". Now when 
the bore is 1/30 of the wave length Bosanquet (see page 425 of the 
volume of Winkelmann above referred to) found a correction for 
the open end of -543 of the radius; and for a bore of 1/12 of the 
wave-length he found '635. For the lowest note C of my flute the 
bore is 1/34 of the wave-length, and I estimate that the correction 
for the open end should be "53 of the radius. Thus I estimate *2o" 



1 This is in agreement with Bosaiiquct's observation, referred to in para. 3 
below, that the correction for the open end of a short tube of a given bore is 
greater than for a long tube. 



The Theory of the Flute. 115 

as the correction for the open end of my flute. The correction for 
the mouth end is thus 2*0" diminished by 0*2", or 1*8"; hence 
that for a half inch hole is 2*4- i*8=o*6"; and that for a f" hole 
is '47. This is for a tube of thickness \" ; for a greater thickness 
the correction would have to be increased. When designing an 
instrument of another internal diameter, say i"* the correction for 
a proportional hole of \" will obviously be o'6"X 1/2=0*3"; for a 
5/i6" hole it will be 0*24". It is however convenient to make 
holes of various sizes; and for holes greater in diameter than "6 of 
the bore the correction can be got by extrapolating from those 
given. But for holes smaller than *6 of the bore other considera- 
tions must be taken into account. 

4. The obvious plan for a concert (lute with a bore of |" would 
be to make all the holes of the same size, say J", not only for the 
notes rising in semitones from D to C-sharp of the bottom octave, 
but also for the bottom C and C-shaip and also for the auxiliary 
topmost holes of the D and TXsharp shake-keys. But if this were 
done the effect of pulling out the tuning slide to flatten the general 
pitch would have a much greater effect when the topmost holes 
were speaking than when the bottom holes were speaking, and the 
instrument would be out of tune with itself. This defect is greatly 
diminished by making the topmost holes smaller than the rest; in 
fact as a hole is made smaller its effect seems to depend more on 
its size and less on its position. In practice the upper C-sharP 
hole, which is perfectly in tune in the lowest octave, is in a position 
which would, if the hole were uniform in size with the lower 
holes, correspond much more nearly to D than to C-sharp. Also 
the D and D-sharp holes, which, like the C-sharp, are only about 
11/32 in diameter, are in positions closely corresponding to the 
wave-lengths of D-sliarp and E respectively : thus in the fingering 
of 6-flat in the third octave the so-called D key is used, as lying 
nearly in the place of Zi-fiat. This same key is used for the finger- 
ing of D-sharp in the fourth octave, while the D- sharp key is 
opened in the usual fingering for E in the fourth octave. 

5. To a beginner in the flute playing the most obvious fact 
in blowing a low note is that as the force of blowing is increased, 
keeping the lips quite still, the pitch rises slightly, and on reaching 
a certain amount of force the note will jump up an octave, then 
becoming usually rather flat in pitch. If the force is further 



n6 G. T. WAJJCER. 

increased the next harmonic may be reached but it also will be 
flat in pitch. After some skill has been acquired the player will 
learn to keep the pitch of a note constant when blowing with 
varying degrees of force by pushing out his lips when blowing 
softly and drawing them in and blowing more downwards when 
blowing \\ith more force. His production of the harmonics will 
then depend on the position and shape of his lips and each 
harmonic can be produced either loudly or softly. The control on 
the pitch of a note that can be exercised by altering the lips is 
much greater when the topmost hole left open by the lingers is 
one of those near the mouth-hole than when it is near the bottom 
end. 

6. Let us now consider the question of fingering, or, in other 
words, the selection of the holes which are to be covered in order 
to produce the successive notes. For an ordinary concert ilute 
in D no question arises over the bottom octave or the middle 
octave as far as A : all that is necessary is to have the ' speaking' 
hole open and one or two holes below it for ' ventilation/ in 
order to produce a free tone. When we are wanting the octave 
above any note, say Z^-flat, we must produce a note which 
instead of having ( nodes ' merely at the mouth-hole and at the 
/J-fiat hole, has also one half-way between (corrections being 
ignored). But for such a wave-length there will also be a node 
at a point whose distance from the mouth-hole is 3/2 times that 
of the #-flat hole. This point corresponds to a note a fifth deeper 
than B-flat. Thus we may produce the note we want by opening 
merely the 5-flat hole and the /i-flat and lower holes: and it is 
easily verified that while with the ordinary lingering it is difficult 
to blow the note up to pitch when a very gentle sound is wanted, 
with this theoretical fingering the gentlest note will not be flat. 1 
Naturally with certain patterns of flutes this method is not appli- 
cable to all notes. For example, a B natural in the middle octave 
cannot be sounded in this way on an ordinary Boehm flute ; for 
covering the F hole will close the hole through which the B 
natural speaks. For the C and C-sharp in the middle octave we 
may define the vibrations in the tube still further by a third node, 



Fingerings of this type are recommended for pianissimo notes in one of the 
foreign hand-books. 



The Theory ol the I* lute. 117 

for the loop at 4/2 times the distance from the mouth -hole is the 
bottom C or C-sharp key. For the latter note we should have 
merely three holes open, the upper C-sharp hole, the F-sharp and 
the bottom C-sharp. In practice however there is no advantage 
over the theoretical method previously described. 

7. For notes in the third octave the methods of cross-linger- 
ing to be found in the text-books differ in several cases from the 
theoretical fingering owing to the mechanical connections between 
certain of the keys. For the theoretical fingering of any note, 
say F-flat, we remember that the distance from the mouth-hole 
to the first node is to be a quarter of that from the mouth-hole to 
that through which the lowest /j-fiat speaks; and we must, as far 
as possible, open the holes corresponding to all the nodes that 
is at 1/4, 2/4, 3/-|, 4/4, 5/4, 6/4, 7/4, etc., of this distance. The cor- 
responding frequencies are 4, 2, 4/3, i, 4/5, 2/3, 4/7, etc. For 
-flat in the third octave, for example, the ordinary fingering 
uses the harmonics corresponding to 4/3 and I ; the Zs-flat key is 
open and that a fourth above this, i.e. the /I -flat key. There is 
no key with frequency 2 on the ordinary Boehm flute, for the 
F-flat shake-key is so small that its position is above that really 
corresponding to Zi-flat for a high note, and if this is opened in 
addition to 4/3 and I the resulting note is too sharp. If however 
the D shake-key is opened instead a good note is obtained. For 
F-sharp the holes to be opened are those producing B, F-sharp 
and D ; and the note so produced is true if the holes are in their 
true places. But if the D-sharp key is, as usual, kept pressed 
down for the sake of rapidity of execution, the /'"-sharp note 
becomes slightly flat; and in order to counteract this error some 
makers put the F-sharp hole in a flute at a trifle higher than its 
true place. The error in the bottom octaves is very slight and is 
not so serious as an error in the top octave where the flute is most 
conspicuous in an orchestra. If the hole for F-sharp is not placed 
above its theoretical position the top F-sharp will be rather flat 
unless the D-sharp key is left closed. For G-sharp the frequencies 
4/3, i, 4/5, 2/3 correspond to a fourth above, G-sharp, a third 
below and a fifth below: or to C-sharp, G-sharp, E y and bottom 
C-sharp. This fingering gives a ringing note that is a trifle sharp ; 
but it is too complicated for ordinary use, and may be simplified 
by opening all the liok'b below the L. We may further simplify 



n8 O. T. WALKER. 

by opening the C natural hole near the top of the tube and thus 
only close the E and F holes in addition to the ordinary fingering 
on the Boehm flute. This will be useful if the ordinary fingering 
gives a flat note. Similarly for the high B-flat we open the holes 
that produce 7>-flat, G-flat, and -flat, leaving the holes below this 
last open. This is rather more complicated than the ordinary 
fingering but is easily played up to pitch while with the ordinary 
lingering flatness is hard to avoid, especially when playing softly. 
The theoretical C-sharp in the topmost octave (with top C-sharp, 
A y F-sharp and bottom C-sharp keys only open) is a trifle sharp. 
But if the top C-sharp key be closed the note is in tune. It 
is however much more complicated than the ordinary fingering, 
and is of no use except when playing pianissimo. 

8. For the few notes of the fourth octave that arc of any 
importance the distances of the holes opened from the mouth-hole 

J'llLir-j 2 



s e" 7 a 3 10 

'O*O^O O O O 

must be proportional to 4/8, 5/8, 6/8, 7/8, 8/8, 9/8, 10/8, etc., and 
the frequencies to 2, 8/5, 4/3, 8/7, i, 8/9, 4/5, etc. For D these 
give D, B-flat, G, E (roughly), D y C y and the ordinary fingering 
for a Boehm flute uses the holes C-sharp (because the D shake-key 
is too highly placed), B-flat, E and C, The other holes are not 
available owing to the mechanical relationships between the keys, 
and perhaps on this account the note is not appreciably sharp. 
For the E of this octave the holes to open are those of top E, C y A y 
F-sharp (roughly) E y D, C. The ordinary Boehm lingering uses 
the D-sharp shake-key, /I, F-sharp, E, C, all other holes being 
closed. In this way the note is difficult to sound; but if the 
theoretical fingering is followed, the C hole is to be opened in 
addition and the lower D hole may also be open, i.e. the so-called 
C-sharp and C-natural keys should not be closed. The sounding 
of the E then becomes fairly easy ; in fact it is easier than the top 
-flat or D. If it is found a trifle sharp the D-bharp shake key 



The Theory of the 



119 



Piawre JZ 

T7;e Tceys OWT* the h-olesfor B <znd 3 c^re 
flute they are not controlled By the fingers directly, but through other keys. 



D 





i 








( 








1 





i 


D 


1 





9 


1 


1 





i 


1 


t 


i 


C* 














o 








o 





o 


C 


1 


1 


1 


e 


1 


1 


1 


I 





I 


B 


' 


& 


/*, 
C^J 


c--^ 





1-3 


,-*, 











B" 








O 











O 











A 


O 











o 





o 


o 








G* 


e 


1 








8 


1 


1 


i 


I 





G 


i? 


^ 
tx 








tj[t* 








e 


e 


? 


F* 








o 








o 


o 


o 


o 


o 


F 





O 








O 

















E 


o 








O 














o 


o 


D* 


i 


t 


e 


s 


s 











t 


1 


D 








o 





o 





o 





o 


o 


C* 





o 


o 


o 


o 


o 





o 


o 


o 


Note 


E 


F 


F" 


G* 


^ 

A 


l^_2 

B k 


C" 


E 


F 41 



The fingeriiys fE- ctnd P are solely of tlieoreticctl interest. For F 
<tnd A closing tfo D A0& JCGSJOS the -note from oeiryflat if there is 
any tencf>n,ry in that direct Inn. The G JiTUterinq rnsty $3 simplified 
7>y openiTy the lower D And D 7u>Zes. Vor & the first fiTtgeri rig is 
strictly theoretical and is inclined to be &tuLrp. So it is better io 
close the temper D key cts? in the second finger ing. For C there Is 
no difficuMy in closing the D Jrey while opcniny ihe D**, hd either \ 
of these alene will suffice. 



T2o O. T. WATJCKR. 

may be closed. In the same way the theoretical F-sharp, with the 
C-sharp, G-sharp, F-sharp, F, D holes open (not the B since opening 
this involves opening the -flat also) may be sounded; but it is too 
difficult to be of practical value. 

9. Some of these fingerings are illustrated in the diagrams 
which arc appended. Figure i is a general scheme for the theoretical 
fingering of any note above the first octave. In it are indicated 
a series of holes a semitone apart of which the open circles re- 
present open holes and the black circles closed : the holes extend 
over a greater range than an ordinary flute contains, but only 
those actually on the flute for the note in question need be con- 
sidered. The numbers immediately above the line are proportional 
to the distances of the holes in question from the mouth-hole; so 
that from 2 to 4 is an octave and from 4 to 8 is an octave. Let 
us now suppose that we want the note which is an octave above 
that corresponding to P t or two octaves above that corresponding 
to (), or three octaves above 7?. The open circles will then obvious- 
ly indicate the holes to be opened, and their intervals above the 
fundamental sounded note will be those given immediately below. 
If, for example, we want the top B which is two octaves above the 
B in the lowest octave, the hole marked 4 must be the B hole ; 
the fourth above is F, so the E hole above this is to be opened if 
there be one. The hole marked 5 is a minor sixth above B y or G ; 
the hole marked 6 is a fourth above or E ; that marked 7 is a 
second or C sharp ; that marked 8 is B which is beyond the range 
of the ordinary flute. On a Boehm flute the hole for 4 is the 
7)-sharp shake-key, while the closing of the holes between 5 and 6 
will close the hole for G, so all holes below the G hole arc left 
open: the note produced is sufficiently controlled by the five holes 
from 4 to 5, and the opening of all holes below the G practically 
cancels the lower part of the tube. 

jo. In Figure 2 are given some fingerings of high notes on a 
Boehm flute, indicated by theory where these differ from the 
standard to be found in the ordinary text-books. The notes in 
question are E, F, F-sharp, G-sharp, /4, 7?-flat and C-sharp in the 
third octave, with E nnd F sharp in the fourth octave. 



X. On Wave-Propagation in Optically Hetero- 
geneous Media* and the Phenomena observed 
in Christiansen's Experiment, 



By Nihal Karan Sethi, M.Sc., Assistant Professor of 
Physics in the Benares Hindu University. 



(Plate VII.) 



CONTENTS. 
I. Introduction. 
II. The Transmitted Light. 
III. The Colours of the Halo. 
IV. The Surface Colours. 
V. Experiments with Liquids. 
VI. The Colons of Doubly Refracting Powders. 
VII. Summary and Conclusion. 

I. INTRODUCTION. 

The very striking and instructive optical experiment known 
under his name was first described by Christiansen in 1884. l He 
found that glass or other isotropic transparent solid in the state of 
fine powder which is ordinarily white and quite opaque, becomes 
transparent in respect of a limited region of the spectrum on 
immersion in a mixture of carbon disulphide and benzol in 
proportions suitable to make its refractive index nearly equal to 
that of the powder. The mixture exhibits a beautiful display of 
colour in the transmitted light and in the halo with which bright 
objects appear to be surrounded when viewed through it. A 
general description of the effects observed will be found in Prof. 
R. W. Wood's Treatise on Physical Optics. 2 It would appear, 

1 Annalen der Physik XXIII, p. 298. 

2 Second edition, p. no. 



122 N. K. SETHI 

however, that in spite of the very attractive features of the 
experiment, the phenomena exhibited by these mixtures have not 
yet been thoroughly investigated. In two important papers, the 
late Lord Rayleigh l drew attention to Christiansen's experiment, 
especially to one remarkable feature observed in it, namely, the 
very small range of wave-lengths transmitted by these mixtures, 
and showed how this featute might reasonably be explained on the 
principles of the wave theory. Apart from this, however, there 
are a number of other important questions which arise and as yet 
remain to be answered. What is the exact distribution of intensity 
of light among the various wave-lengths transmitted ? How does 
this distribution depend on the size of the particles in the powder 
and the length of path of light through the mixture ? Has the 
halo a definite structure, or must we be content with vaguely 
describing it as having a colour complementary to that of the 
transmitted light? How is the halo influenced by the factors 
mentioned above and the other circumstances of the experiment ? 
It is proposed in the present paper to examine these and other 
questions relating to the propagation of light in optically hetero- 
geneous mixtures. It is believed that the inquiry may be of 
interest, especially in view of the very wide application in recent 
years of the method of immersion in the determination of refractive 
indices and dispersive powers. 

II. THE TRANSMITTED LIGHT. 

According to the principles of geometrical optics, a mixture 
of two isotropic media should regularly transmit only the rays for 
which the refractive indices of its components are identical, and 
should scatter all others in various directions. Actually however, 
in Christiansen's experiment, we find that this is not the case and 
that part of the incident light is regularly transmitted and is 
capable of giving well-defined optical images even when the 
refractive indices differ by an appreciable quantity. For instance, 
if a flat-sided cell containing the mixture be placed between the 
collirnator and the prism of a spectroscope of which the slit is 
illuminated by sunlight, we find that a finite region of the spectrum 
continues to be visible and is in sharp focus, the Fraunhofer lines 

1 Scientific Papers, Vol. II, p. 433 aud Vol. IV, p. 392. 



On Wave-Propagation in Optically Heterogeneous Media. 123 

being quite clearly distinguishable in every part of it. The range 
of wave-lengths thus regularly transmitted in the spectrum varies 
with the fineness of the particles and the thickness of the layer of 
powder. Christiansen in particular found that the transmitted 
region increased in width with the fineness of the powder used and 
indeed comprised nearly the whole visible spectrum when the 
observations were made with the finest powders of all separated by 
prolonged elutriation. The explanation of these effects is obviously 
of considerable interest. 

The late Lord Rayleigh * proposed a treatment on the following 
lines. He first determined the probability that the number of 
particles of one component which a ray will encounter during its 
passage through the mixture, will differ from the mean number 
m by less than r, and showed that this will be sufficiently great 
and equal to '84 for a value of r=\/2m. This gives a measure of 
the phase differences likely to arise in the passage of the light 
through the heterogeneous medium, from which it is inferred that 
the range of wave-lengths transmitted may be expected to be 

, times that which is not resolved by a prism of equal thickness 
v 2m 

with a dispersive power equal to the difference between the dis- 
persions of the two media. The expression for the latter resolving 
power is well known to be 



'z 

and even if m is taken to be equal to -- a value evidently 

a 

greater than the real one the total range of wave-lengths trans- 
mitted by the powder should be 



. p 
d\ 

--~ (i) 

\/2td * 



Scientific Papers, Vol. IV, p. 395. 



124 N - K. SETHI. 

where t is the thickness of the layer of the medium and d is the 
average size of the particles in it. 

It would appear, however, from the investigation made by 
the present writer that the observable width of the region of trans- 
mission in the spectrum depends to a large extent on the intensity 
of light used and in all the measurements made, it was found to 
be much greater than that indicated by Rayleigh's formula. Even 
Christiansen's estimate is much greater than what equation (i) will 
give for his data. The fact is that the distribution of intensity 
among the wave-lengths transmitted is of the exponential form, 
being maximum for /*-//=n0 and diminishing more or less rapidly 
to zero for large values of /*-/*/ (M and /*' are the refractive indices 
of the two components of the mixture). We cannot, therefore, 
strictly speaking, talk of a region of transmission without at the 
same time defining the limiting intensity of the light at the ends 
of that region. 

Fortunately, considerations of the wave-theory of light com- 
bined with the theory of probability readily give us an expression 
for the intensity of the transmitted light, We may treat this 
problem on much the same lines as adopted in a recent paper by 
Mr, Chinmayanandam l in discussing the specular reflection from 
a rough surface. Consider the configuration of the wave-front 
immediately on emergence from the heterogeneous medium. It 
will not be a perfectly plane wave-front, because different portions 
of it will have been unequally retarded by the random distribution 
of the particles of the powder. We may assume that the deviations 
of the wave-front from perfect planeness follow the well-known 
law of errors. Certain parts of the wave-front would have traversed 
more of the solid and less of the liquid and vice versa. lyet x 
denote the difference between the average thickness of the solid 
traversed by the whole wave-front and the thickness traversed by 
a specified element of the wave-front. Then the total area of the 
portions of the wave-front in advance of or behind the mean, for 
which this difference lies between % and x-\-d% 9 may be taken 

A tr2 

proportional to e ' dx. The resultant vibration due to these 
portions of the wave-front is, therefore, 



Physical Review, Vol, XIII, Feb. 1919, p. 96. 



On Wave-Propagation in Optically Heterogeneous Media. 125 

y = e~~ x cos \ wt (/A //)# > dx , 

if the equation of the vibration for the mean wave-front be y= 
cos wt. The average vibration due to a unit area of the complete 
wave-front will then be for the wave-length A , 



!. 



Ax* ( J 27T, ,. 

c cos< wt (n n) 



-AS 



This will make the average intensity 

, -?() -()' 

/ e =6' (2) 

where the constant B will depend on the size of the particles d and 
the thickness of the layer of powder t and possibly to sonic extent 
on the arrangement of the particles. 

To test the foregoing formula and to find experimentally the 
manner in which the constant B depends on d and t, a number of 
observations were made by the writer. Powdered glass was the sub- 
stance used being the most convenient and suitable for the purpose. 
By passing it in succession through several sieves of wire gauze and 
cloth having a different number of meshes to the inch, powders of 
varying grades of fineness were obtained, the particles in each of 
which lay between definite and determinable limits of size. The 
samples of powder thus obtained were then thoroughly cleaned, 
washed and dried. For the purpose of the observations, the 
powder was placed in a cell with parallel faces of optically good glass, 
and enough carbon disulphide was poured into the vessel. Benzene 
was then slowly added till the refractive index n of the glass and 
/*' of the liquid mixture became equal for red light and this began to 
be freely transmitted. Further additions of benzene shifted the 
region of transmission towards the violet end of the spectrum, thus 
affording opportunity for making observations at any desired stage. 
The thickness t of the layer through which the light had to pass 
was varied by inserting into the powder pieces of clean glass plate 
of different thicknesses. 



126 



N. K. SETHI. 



The constant B was determined by placing the cell containing 
the powder between the collimator and the prism of a wave-length 
spectroscope and determining the range of wave-lengths trans- 
mitted, then screening the light by a screen which transmitted a 
known fraction k of the light falling on it, and then again deter- 
mining the diminished range of tiansmission. The source of light 
must be quite steady and a 500 c.p. half-watt lamp was used. It 
is clear that if the limit to which the eye of a particular observer 
in particular circumstances can just see, corresponds to a certain 
absolute value of intensity, the value of the intensity at the limit 
in the second case after screening must be equal to that at the 
limit in the first case. So that 



- 



ke 



and 



B= -- 



log* 



v //* 

I ~v 



(3) 



For comparison three such screens were employed for which the 
values of k as determined by a rotating sector photometer were 

, and respectively, and the values of B determined 

8-41 13-44 22.5 

by each of them separately agreed well. The variation of B with 
t and d is set forth in tables I and II. 

TABUS I. 

^=0055 c.m. 



t in c.m 



B 





T 



-0528 



0480 



7 8 



0315 



0404 



45 



0161 



0358 



30 



0130 



0433 



00705 



0588 



On Wave-Propagation in Optically Heterogeneous Media. 127 



TABLE II. 


^ = 78 c.m. 


d in c.m. 


D 


B 

d 


0055 


0315 


5-69 
3-63 


0142 


0515 


0245 


0900 


3-67 


0410 


206 


5-03 



In view of the unavoidable uncertainty in the observations 
and changes in the temperature of the mixture and also the non- 
constant nature of the arrangement of the particles, the values of 

r> D 

and - may be seen to be fairly constant for the large variations 
t d 

of the values of t and d. So that B must be taken to be of the form 



and / = (4) 

where c is a constant which may depend on the arrangement of 
the particles but is otherwise an absolute one. It may be noted 
that this formula makes the dimensions of c as of a pure number. 

The result B=ctd may also be deduced theoretically if we 
combine (i) and (2). If we remember that 



at the limit of visibility, we get 



i 

'8td 

i 
: 8td' 



(5) 



Now, if we suppose that at the limit given by Rayleigh's formula, 
the intensity falls down to a fixed value, we must have from (2) 



128 N. K. SETHI. 

I 

= constant 

V A / 

from which (5) will give 

B 

- - = constant 

or B = ctd. 

The average value of the absolute constant c as deduced from 
the experiments described above is about 7. 

From the expressions given above, it is a simple matter to 
calculate the proportion of the energy that appears in the trans- 
mitted light, given the size of the particles, the thickness of the 
layer and the dispersive powers of the media or vice versa. It 
would be interesting to continue the work and test experimentally 
whether the formula given by the foregoing theory continues to be 
valid in the case of powders much finer than those used by the 
author in his observations and which transmit a much larger 
portion of the spectrum. No quantitative estimate is given by 
Christiansen of the size of the particles in the case of the finest 
powders used in his observations which according to him transmit 
nearly the whole of the visible spectrum. Otherwise this question 
could very easily have been tested. It seems not unlikely that 
the opalescent coloured precipitates of potassium fluosilicate ex- 
perimented upon by Wood may be usefully employed for further 
work on this point. 

III. THE COLOURS OF THE HALO. 

The energy which fails to appear in the transmitted light goes 
mostly into the halo seen surrounding the light-source, the propor- 
tion lost by reflection or absorption being small except possibly in 
the case of large thicknesses. Some observations have been made 
by the writer of the manner in which the scattered light is 
distributed in the halo. A strong source of light an arc-lamp or a 
1000 c.p. half watt lamp was placed behind a small hole which 
was observed through the mixture of the powder and the liquids 
from a distance of about two meters. Naturally, the source 
appeared to have the colour of the transmitted light and was 
surrounded by a halo which was also coloured and which extended 
to very large angles. Hitherto the halo has been described as 



On Wave-Propagation in Optically Heterogeneous Media. 129 



having a colour complementary to that of the transmitted light. 
This suggests that the halo is all of one colour. But the most 
casual observation was enough to show that this was not the case 
and that the colour of the halo changes progressively from the 
centre outwards. The part of the halo immediately surrounding 
the source is of nearly the same tint as the source itself and shows 
a fine mottled or granular structure similar to that exhibited by 
diffraction halos in monochromatic light which has been discussed 
by De Haas l in a recent paper. The following table shows the 
colours of the different parts of the halo in various cases, the order 
being from the centre outwards : 

TABLE III. 



TRANSMISSION. 



HALO. 



Red 
Yellow . . 



.. i Red, Orange, Yellow, Greenish-yellow, Green, Blue, Indigo. 



Yellow, Orange, Orange-yellow, Yellow, Yellowish-green, Green, 
Greenish-blue, Blue, Indigo. 



Green . . . . Green, Orange-yellow, Rose, Lilac. 



Blue 



Indigo . . 



Violet 



Blue, Greenish-yellow, Orange-yellow, Rose, Red. 



Indigo, Blue, Bluish-white, Greenish- white, Yellowish- white, 
Reddish-white. 



Violet, Blue, Bluish- white, Greenish- white, Yellowish- white, 
Reddish-white. 



The explanation of the variation of the colour of the halo in 
its different parts becomes evident on examining the spectrum of 
the light scattered by the powder. In order to do this, all that is 
necessary is to take a direct-vision spectroscope and point it towards 
the cell containing the mixture. But the light proceeding in 
directions other than the one desired must be prevented from 
falling on the slit of the spectroscope. This can be ensured if the 
distance between the powder and the spectroscope slit is sufficiently 



1 K. Akad. Amsterdam Proc. XX, pp. 1278-1288, 1918. 



130 N. K. SETHI. 

great (over a ineter or two) and the area of the surface of the 
powder sending out the light to the slit is itself considerably 
restricted by a narrow aperture. Specially interesting are the 
effects observed at very small angles. Fig. I in the plate shows 
the spectrum in the direction of the freely transmitted light and 
consists of a very narrow band for which M-M' does not appreciably 
differ from zero. Figs. 2-5 in the plate are the spectra at gradually 
increasing obliquities. Fig. 6 is the complete spectrum reproduced 
for comparison of the widths of the bands obtained in the various 
cases. From these it is clear that proceeding from the centre, the 
halo consists at first mainly of two very narrow bands in the 
spectrum on either side of the region of transmission; these bands 
gradually widen out and also become more and more separated as 
the obliquity of observation is increased. The actual colour of 
the halo in any case is the resultant due to the mixture of the two 
bands in the spectrum. At a sufficiently great obliquity, these 
bands will together comprise almost the whole of the spectrum 
except for the wide region included between them in which the 
intensity is deficient or actually zero. In this case the colour of 
the halo may be said to be roughly the complementary of that of 
the transmitted light roughly because a considerable part of the 
spectrum is missing from it over and above the portion regularly 
transmitted through the powder. 

Further information regarding the halo is obtained by using a 
monochromatic light-sourcesay the green light of the mercury- 
vapour lamp and varying /*-/*' for this light by altering the 
proportions of carbon disulphide and benzene in the mixture. But 
a better and more convenient method is to keep the mixture un- 
altered and change the wave-length of the light used. A mono- 
chromator must be used for this purpose. It is most instructive to 
observe, as the screw of the instrument is slowly turned, the halo 
decrease in size, vanish altogether the intensity of the regularly 
transmitted light being at this instant a maximum and then 
increase once again. From the general appearance of the mono- 
chromatic halo and some photometric observations made by the 
author, it would appear that the distribution of light in it is at 
least approximately of the form e <***, the illumination being 
maximum at the centre and gradually diminishing symmetrically to 
zero. The constant a must also decrease with increasing M-M' and 



N. K SETHI PLATE VII, 



SPECTRA 



Fig I 2 Transmission 



FIR 4 



g. 5 



I 


S c a 1 1 e r i n K 


1 


Ditto 
at greater obllqu Ity 


I) 





Complete Spectrum 



Scattering by 

F| 6 V - ' :: -^ r Glass 



C^artz 



Phenomena observed In Christiansen's Experiment. 



On Wave-Propagation in Optically Heterogeneous Media. 131 

very rapidly too ; for the extent of the halo, as far as the eye can 
see it, is very small at first and increases very rapidly as ^-/^ 
increases. It is also clear that it should be an even function of 
M /*' so that at least as a first approximation a may be taken to be 

of the from , where 6 depends on the size of the particles d y 

(M-/O* 

the thickness of the layer of powder t and possibly on the wave- 
length A. 

In fig. 7 are drawn a few curves of the form y=e- a ** for 
different values of a and correspond to the illumination curves for 
different wave-lengths. The abscissae represent the distances from 




the centre of the halo. Curves i, 2, 3, 4, correspond to cases in 
which the difference /*--/*' has steadily increasing values. It is 
at once apparent from the figure that at the point A of the halo, A, 
corresponding to curve i is practically absent and the principal 
illumination is due to wave-lengths near about A. 2 and A/ (one 
greater and the other smaller than A,) corresponding to curve 2. 
Similarly at />, even A 2 and A/ arc almost absent and the main 
illumination is due to wave-lengths in the neighbourhood of A 3 
and A 3 '. This completely agrees with the photographs in figs. 
2-5 and satisfactorily explains why the dark band in the centre 
and the two bright bands on either side gradually widen out as we 
proceed further and further out from the centre of the halo. 



132 N. K. SETHI. 

The writer is not yet in a position to elucidate the exact 
manner in which the numerical value of the constant b determining 
the angular size of the halo depends on d, t and A, or to offer a 
detailed theory. The general nature of the variations was, 
however, evident from the observations, namely that the halo 
increases in size when the particles in the mixture are smaller or 
the thickness of the layer is larger. Much more photometric work 
is necessary before this question and the manner in which A comes 
in can be definitely settled. Reference may, however, be made 
here to the explanation of the granular structure of the central 
part of the halo which has already been mentioned. This is 
evidently connected with the arbitrary variations in the phase at 
different points of the wave-front emerging from the mixture. 
Any quantitative theory of the distribution of light in the halo 
can give only the statistical or average value of the intensity in 
the neighbourhood of any given direction. At individual points of 
the halo, large deviations from this average value arc inevitable. 
Near the centre of the halo, as we have already seen, the scattered 
light is confined to a very small region of the spectrum, and 
hence the arbitrary fluctuations of intensity are specially marked. 
Further out from the centre of the halo, the effects of different 
parts are superposed and the irregular fluctuations of intensity are 
less easily noticeable. 

That the size of the halo surrounding the source varies with 
the difference /< /'' of the refractive indices and hence is different 
for different colours of the spectrum is very clearly illustrated by 
fig. 8 in the plate. This is a picture obtained by interposing the 
cell containing a uniform layer of the mixture in the path of the 
rays forming a focussed spectrum on the photographic plate, the 
distance between the plate and the cell being about half a centi- 
meter. Owing to the scattering action of the layer, the different 
parts of the spectrum are spread out to different extents, this 
spreading being evanescent for the point at which /*--/*/ There is 
a distinct asymmetry in the photograph which would seem to 
indicate that the size of the halo is greater on the large wave-length 
side and less on the other. 

IV. THE SURFACE COLOURS. 
A very striking effect which does not appear to have been 



On Wave-Propagation in Optically Heterogeneous Media. 133 



previously described is the brilliant colour of the boundary between 
the glass powder in the lower part of the vessel and the liquid 
above it. This colour is different from the rest of the powder and 
is ever so much more brilliant. In order to observe it well, it is 
necessary that the surface of the powder should be level, which 
can be easily secured by gently tapping the vessel, and the line of 
sight should be almost tangential to this surface. Another very 
remarkable thing about this boundary is that it does not show the 
same colour on both sides, so that when regarded from the lower 
or the glass side, it shines with one colour which changes into quite 
a different one when the observation is made from the upper or 
the liquid side. Both these appearances are to some extent 
summarised for the various stages in the following table in which 
the order of the colours is for gradually increasing angles with 
respect to the transmitted light, the most beautiful stage being 
when the yellow is freely transmitted. 



I TRANSMISSION. 

Red. 
Yellow. 
Green. 
Blue. 
Violet. 



TABUS IV. 

SIDE. 



Green, Blue. 



Green, Blue, Violet. 



POWDER SIDE. 



Deep Red. 



Orange, Red. 



Blue, Violet. Yellow, Orange, Red. ; 



Violet. 
Faint Violot. 



Green, Yellow. 



Groon, Yellow. 



The reason for these colours appears to be that the surface 
of separation acts more or less like a totally reflecting barrier, 
those wave-lengths for which M>M' being totally reflected on the 
glass side, while those for which /'.</ appear on the liquid side. 

The fact that the boundary colours appear brilliant when the 
line of sight is almost tangential to the surface is easily explained 
if we remember that the difference of refractive indices is at most 
very small and hence the angle of incidence should not differ much 
from a right angle. 



134 N - K - SETHI. 

These colour effects are identical with coarse as well as fine 
powders, the only difference being that with fine powders the 
boundary line appears much sharper and more well defined. 

V. EXPERIMENTS WITH LIQUIDS. 

The colours obtained when equal volumes of glycerine and 
turpentine arc shaken together have been noticed previously and 
Wood remarks that they are of a similar nature to those observed 
in Christiansen's experiment. It was thought that it would be 
worth while to examine this case more closely. At room tempera- 
tures, pure glycerine has a refractive index slightly less than that 
of spirits of turpentine throughout the visible spectrum, the 
difference being greatest at the violet end and least at the red. The 
effects noticed at ordinary temperatures are not therefore parti- 
cularly striking. The author has found however that at higher 
temperatures very lively colours may be obtained. The liquids 
may be put together in a small flat-sided flask which is corked and 
then warmed up by being put in a beaker containing boiling water 
for a few minutes. On taking it out and shaking up the mixture 
and looking through it at a source of light, we find no transmission 
at first but merely a halo with violet centre surrounded by the 
other spectrum colours in regular order. As the mixture cools, 
violet, blue, green, yellow and red light is transmitted in succession 
and the colour of the halo undergoes corresponding changes. 
Finally, the transmitted light vanishes, and we have only a halo 
with a yellowish white centre and a blue-violet margin. The size 
of the halo in all these cases is much smaller than in the case of the 
glass powder and consequently it is much easiei to distinguish the 
colour of the different parts of the halo. Another interesting point 
about this halo is that the mere shaking up of the mixture increases 
its size as one should naturally expect from the greatly diminished 
size of the drops. 

The upper liquid, viz. turpentine, becomes very quickly free of 
the large drops of glycerine which fall down rather rapidly, but 
very fine drops remain floating in it for a considerable time. When 
an object is viewed through this apparently clear liquid, a halo is 
seen to surround it, while the object itself remains practically white. 
This halo is quite different from the halo referred to above and 
which is seen through the lower liquid where one can see many 



On Wave-Propagation in Optically Heterogeneous Media. 

large drops. It is much smaller in size and does not show much 
gradation of colour. Generally only one or two tints are visible. 
When the transmission is near the red end of the spectrum, this halo 
consists of blue and violet colours only. When the transmission is 
towards the other end, the halo is brown and red. In the 
intermediate stages it has a purple or a pink colour. Both this 
and the proper halo show a very curious appearance of activity 
near their centres which strongly reminds one of the appearance in 
a spinthariscope. This is no doubt due to the drops slowly falling 
down through the liquid. 

Looking at the surrounding objects in daylight through the 
hot mixture, the general transparency is found to be very great, the 
range of spectral transmission being rather considerable. Observed 
with a direct vision spectroscope, the transmitted light appears 
not as a sharp band as in fig. I in the plate but much wider. 
Similarly a spectrum seen through this mixture when hot shows a 
much broader region in good focus than does the glass powder. 
This is 110 doubt due to the smaller difference in the dispersive 
powers of the constituents of the mixture. The same is true of 
the halo also. The dark bands corresponding to figs. 2-5 are 
obtained but they are never so narrow. 

As the small drops fall through the liquids, something like a 
boundary of separation forms between the clear turpentine above 
and the collection of drops below. This boundary like the one in 
the case of the glass powder shows brilliant colours different from 
the rest of the mixture and also different on the two sides. When, 
however, the drops become large, there is nothing like a regular 
boundary and no such effects are produced. Individual drops no 
doubt produce certain diffraction effects and their edges shine out 
with various colours depending on the difference of refractive 
indices of the two liquids and the angle of observation. These 
peculiar effects are visible not only in the drops just below the 
clear turpentine, but also in practically all other drops. This gives 
rise to a sparkling appearance in the liquid and is particularly 
noticeable when the mixture is just taken out of boiling water and 
its temperature is rather high. These eifects will be discussed in 
a separate paper. 

It may be worth while also to record the results observed 
when glycerine is shaken up with rather a small quantity of 



136 N. K. SETHI. 

turpentine. In this case, a milky emulsion of the two liquids 
results, which takes a very long time to clear and which may, 
therefore, be very conveniently used for observations of the effect 
of warming on its transparency. Partly on account of the very 
fine state of subdivision in the mixture thus obtained and partly 
no doubt also on account of the small quantity of turpentine taken, 
up by the glycerine, the region in the spectrum transmitted by it 
when warmed up is very great in this case. 

VI. THK COLOURS OF DOUBLY REFRACTING POWDERS. 

Christiansen mentions in his paper that doubly refracting 
powders do not give satisfactory results in his experiment, as the 
colours they show are not so pure, and the mixture is never fully 
transparent. On the other hand, we find that Prof. R. W. Wood 
remarks that the best results that he could obtain were with 
powdered quartz. It is difficult to reconcile these contradictory 
statements, for although it may be true that so far as the general 
appearance of the powder and the halo is concerned, powdered 
quartz does perhaps give more attractive and lively colours than 
does glass and the halo is also much wider, yet there does not 
remain the slightest doubt after a closer examination of the pheno- 
mena that Christiansen was absolutely correct in his statement 
referred to above. We find in fact that in the case of quartz the 
whole of the light that emerges through the powder appears in the 
halo and there is, strictly speaking, no light regularly transmitted 
and capable of forming a definite image of the source. The real 
Christiansen phenomenon is as a matter of fact altogether absent. 
This is apparent on observing a small brilliant source the filament 
of an electric lamp or a narrow aperture backed by an arc lamp 
through a layer of the mixture containing the quartz powder, 
when it will be found that even with very small thicknesses used, 
there is no defined image of the source. It remains invisible and 
we merely see a halo surrounding its position the colours of which 
gradually alter as we proceed from the centre outwards. It would 
therefore seem that what Prof. Wood calls the *' transmitted 
colour " was merely the central portion of the halo and he could 
not have seen the actual image of the lamp flame. 

The spectroscopic examination of the phenomenon in the 
manner described in the preceding sections makes this all the 



On Wave-Propagation in Optically Heterogeneous Media. 137 

more clear. None of the appearances shown in figs. 1-5 in the plate 
can be observed with quartz powder. Even in the direction of 
the incident light almost the complete spectrum is visible with 
those portions of it somewhat more intense for which the refractive 
index of the liquid approaches that of quartz. In directions more 
and more oblique to the incident light, this brighter portion 
gradually loses, while the portions on either side of it gain in 
intensity, but the difference between them becomes appreciable 
only at rather large angles. The colours observed either in the 
direction of the incMent light or far out in the halo are therefore 
all impure by the admixture of a fair quantity of white light. 

When the cell containing the mixture of powdered quartz was 
placed between the collimator and the prism of the spectroscope, 
no portion of the spectrum could be seen. There was only a 
general illumination of the Held. 

The complete absence of the transmitted light is, however, 
best illustrated by fig. 9 in the plate which is the ordinary spectrum 
as seen through the quartz powder. Fie. cS is the spectrum seen 
through the glass powder under similar circumstances It will be 
noticed that while in iig. 8 there is a certain region of the spectrum 
which appears to be well focusscd and free from halo, there is no 
such region in fig. 9. The halo does no doubt attain a minimum 
size, but it does not vanish ut any place. And this happens even 
for very small thicknesses of the powder. 

This result may appear to be surprising at first sight, but is 
easily explained. For, on account of the doubly refracting nature 
of the quartz, and the absolutely random orientation of the particles 
with respect to their crystallographic axes, no ray can travel through 
the mixture from start to finish with one definite velocity. When- 
ever it will encounter a particle of quartz, it will in general be broken 
up into two, travelling with different velocities and in some cases 
when although it is not broken up into two, an ordinary ray may be 
wholly transmitted as an extraordinary ray. So that every ray must 
be regarded as partly ordinary and partly extraordinary during its 
passage through the powder. No possible value of the refractive 
index of the liquid mixture can allow such a ray to pass as through 
a homogeneous medium. 

The question then arises, fl To what value of p does the 
minimum size of the halo in fig. 9 correspond ? " In order to settle 



138 N. K. SETHI. 

this, a wave-length spectroscope with a shutter eye-piece was used. 
The shutter was closed down to a narrow slit so that the light 
passing through it had a definitely known wave-length which 
could be quickly altered. With such an arrangement the wave- 
length corresponding to the? minimum halo could be quite accurately 
determined. 

A suitable refractometer not being available, the refractive 
index of the liquid corresponding to the minimum halo was found 
out by a rather indirect but quite a simple method. A tiny little 
prism of the same quartz (one of the broken pieces) was suspended 
in the liquid above the powder. Looking through this, one could 
sec two images of the slit in the shutter eye-piece. It was easy to 
find which of them corresponded to the ordinary ray, for in the first 
place MO is smaller than p> P , and secondly the ordinary image did 
not move as the prism was turned, while the other moved so that 
the distance between them altered. By turning the wave length 
screw, it was easy to make the ordinary image coincide with the 
slit as seen directly from above and below the prism. For this 
value of A, / A '=M(). From the known values of the refractive 
indices of carbon disulphide and benzol for this wave-length the 
proportion of the two liquids in the mixture was calculated l so as 
to satisfy the condition A'=/'O. The proportion having been found, 
the /<' for the wave-length corresponding to the minimum halo 
could be calculated. A specimen result is quoted below : 

minimum halo at /x i 

/<'=/x at A & 

But /* =1-5454 at A 2 

/x / =i-5454 at A; 
and =1-5488 at \. 

Theoretically one would expect on an average half the path 
of a ray through the powder to be ordinary and half extraordinary. 
So that if /*' is equal to the mean of the ordinary and the mean 
extraordinary indices, or in other words if 



See Prestion : Theory of Ivight, pa#e 134. 



On Wave-Propagation in Optically Heterogenous Media. 139 

the path difference between a ray passing through the liquid and 
one through the quartz may be expected to be the least. This value 
of /<' should, therefore, give rise to the minimum halo. Now, for 



=5270, MO= i'545 a 

'555*) { ==1-54865 
in complete agreement with the value obtained experimentally. 

VII. SUMMARY AND CONCLUSION. 

The paper describes the results of a detailed study, both 
experimental and theoretical which has been made of the pheno- 
mena observed in Christiansen's experiment in which the powder 
of a transparent substance is immersed in a mixture of carbon 
disulphide and benzol having a refractive index nearly equal to 
that of the powder. 

(i) Transmitted Light. 

(a) The observable range of wave-lengths transmitted by the 

mixture has been found to depend on the intensity of 
the incident light. 

(b) A theoretical treatment based on the principles of the 

wave-theory and the theory of probability has been 
given, and it shows that the intensity of the transmit- 
ted light is given by the expression 



-cidf 1 *^} 
I=c V * ' 



The influence of the size of the particles and the thick- 
ness of the medium on the intensity of the transmitted 
light has been confirmed experimentally and the value 
of the absolute constant c has been determined and 
found to be roughly equal to about 7. 

(r) The formula shows that with the very finest powders, the 
range of transmission may be very considerable. 

(2) The Colours of the Halo. 

The statement made hitherto that the colour of the halo is 
complementary of that of the transmitted light is not correct. The 



140 N. K. SETHI. 

halo is not of one colour throughout, but has a definite structure. 
This is shown by observations and photographs of the spectrum of 
the halo, which at small obliquities consists of two narrow bright 
bands separated by a dark interval. These bands widen out as 
the angle of observation is increased. The facts are explained by 
observations in monochromatic light which indicate that the 
distribution of intensity in the halo may be taken to be of the form 



which is of the exponential type, where b increases as the size of 
the particles is increased and as the thickness of the layer or the 
wave-length of light is diminished. 

(3) The Surface Colours. 

The level surface of separation between the clear liquid on the 
top and the powder below exhibits remarkably brilliant colours 
which arc not only different at different stages of the mixture as 
regards the refractive indices, but are also different on the two sides 
of the boundary. This effect does not appear to have been noticed 
so far. Presumably it is due to a sort of total reflection from the 
boundary. 

(4) Liquid Mixtures. 

A mixture oi glycerine and turpentine shows similar pheno- 
mena, but these become evident only when the llask containing the 
mixturr is heated by immersion in hot water. At ordinary 
temperatures there is no light transmitted. The halo in this case 
is, however, much smaller, but the general transparency of the 
mixture is very great. Some interesting diffraction effects by the 
edges of the liquid drops are also observed. 

(5) The Colours of Doubly Refracting Powders. 

Doubly refracting powders, such as quartz, are, contrary to 
the statement of Prof. R. \V. Wood, unsuitable for the exhibition of 
the true Christiansen phenomenon, because in their case there is 
no regularly transmitted light. The whole of the light emerging 
through the powder appears in the halo, which is differently 
coloured in its different parts. There is a particular wave-length 



On Wave-Propagation in Optically Heterogeneous Media. 141 

for which the size of the halo is a minimum and this has 
been shown both experimentally and theoretically to correspond 
to the case in which the refractive index of the liquid is equal to 



In conclusion the writer has much pleasure in recording his 
indebtedness to Prof. C. V. Raman who took considerable interest 
throughout the progress of the work and whose suggestions and 
criticisms were of immense help. The experimental work wa v 
carried out at the Laboratory of the Indian Association for the 
Cutlivation of Science. 

Dated Calcutta, 
The 2nd of November 1920. 



XL On the Production of Musical Sounds 
from Heated Metals. 



By B. N. Chuckerbutti, M,Sc. f Assistant Professor of 
Physics, Calcutta University. 



(Plates VIII to X.) 

CONTENTS. 
I. Introduction. 

II. The Vibration-Curves of the Trevelyan Rocker. 
III. Description of the Observed Phenomena. 
IV. Crucial Test of Davis's Theory. 
V. Summary and Conclusion. 

I. INTRODUCTION. 

That vibrations giving rise to musical sounds may be produced 
under suitable conditions by the contact of metals at different 
temperatures, has long been known as a fact of observation and is 
best illustrated by the familiar piece of apparatus known as 
Trevelyan's Rocker. As is well known, this apparatus usually 
consists of a bar of metal (generally brass) prismatic in form which 
has a narrow longitudinal groove cut along its lower edge, so that 
if it be placed on a table, the metal rests on two adjacent parallel 
ridges. If displaced from its position of equilibrium, the bar rocks 
to and fro, resting alternately on the two lines of support, oscillat- 
ing at first slowly and then more quickly as the amplitude dies 
away, the energy being gradually dissipated in the succession of 
impacts of the points of support upon the table. In actual use the 
rocker has a handle which consists of a brass rod with a ball at one 
end. The rocker is heated and is then supported in a slanting 
position with the edges of the groove touching the clean cold surface 
of a horizontal block of lead, the end of the handle resting on the 



144 B. N. CHUCKERBUTTI. 

table. The rocker has generally to be started, that is, one edge 
has to be lifted out of contact with the lead block and then allowed 
to come down. The movement then continues to sustain itself, a 
rough sound being heard at first which later gives place to a tone 
of markedly musical character. 

The generally accepted explanation of the working of the 
Trevelyan rocker (due in the first instance to Sir John Leslie) was 
developed by Faraday. 1 The correctness of Faraday's explanation 
was questioned by Forbes,* but was supported by Seebeck* and 
later by Tyndall,* who carried out an extensive series of studies on 
the subject. An experiment illustrating Faraday's theory was also 
made by Page 6 in which the preliminary heating of the rocker is 
dispensed with and replaced by the local heating at the points of 
contact produced by the passage of an electric current during the 
experiment. Briefly stated, the view now generally held regarding 
the mode of action of the apparatus is as follows : 

(1) The rocker moves under the action of gravity (its own 
weight), resting alternately upon two ridges of support. 

(2) This movement is sustained by the local periodic expan- 
sions of the lead block (which has a high rate of expansibility with 
increasing temperature and a low heat-conductivity), in consequence 
of the intermittant contact with the heated metal. The forces 
brought into play by these expansions do work and supply the 
energy requisite for the maintenance of the motion. Davis* 
examined the subject mathematically in a paper which we shall 
have occasion to refer to again, and came to the conclusion that 
the foregoing explanation of the maintenance of the motion was 
adequate. 

The present work was undertaken with a view to determine 
by direct experimental study (and not merely on a priori supposi- 
tions) the manner in which the heated bar vibrates and gives rise 
to musical tones when placed in contact with the cold metal. The 



J Faraday : Experimental Researches iu Chemistry and Physics, page 311. 

2 Forbes: Phil. Mag., Vol. IV, pp. 15, 182, 1834. 

3 Seebtck : Pogg. Ann., Vol LI, p. 1, 1840. 

* Tyndall: Sound, third edition, page 52, ed. 1875; phi1 ' Trans. Roy. Soc., 
Parts I-II, 1854. 

* Page: Silliman's Journal, p. 105, 1850. 

6 Davis: Phil. Mag., Vol. XLV, p. 296, 1873. 



B. N. CHUCKERBUTT1. 



PLATE vin. 




Fig. I 



Trevelyan Rocker arranged to photograph Vibration Curves, 




Fig. 2 



Fig. 3 



Fig 4 



Vibration Curves showing effect of Asymmetric Pressure. 



Musical Sounds from Heated Metals. 145 

outcome of the research has been to show that the view usually 
held regarding the mode of action of the apparatus requires to be 
modified in essential particulars. 

II. TlIK VIBRATION-CURVKS OF THE TREVEI.YAN ROCKER. 

The motion of the rocker in the later stages of the experiment, 
when it gives rise to musical tones, is of quite a small amplitude 
and requires to be highly magnified to be satisfactorily observed or 
photographed. The method adopted by the writer has proved 
itself to be exceedingly simple and effective in practice and is 
shown in fig. I (Plate VIII). A sewing needle is placed horizon- 
tally on the curved upper surface of a small metal block which is 
firmly attached to the rocker. When the rocker vibrates, the 
needle rolls between this surface and a small bar of wood which is 
placed lightly resting on the upper surface of the needle, the other 
end of the bar being supported at a suitable level above the 
surface of the table. The rotation of the needle is indicated by a 
mirror attached to it, on which is incident a pencil of light from a 
pin-hole illuminated by the electric arc. The light reflected from 
the mirror is focussed by a lens upon a moving photographic plate 
or by a suitable arrangement can be projected upon a screen to 
make the form of the vibration-curves visible. Magnification of 
the movement of the rocker ranging from l,ooo to 20,000 times may 
be readily obtained in this way. 

By fixing the metal block on which the needle rolls firmly to 
the rocker itself, the motion of the later is readily observed. On 
the other hand, the block may be fixed, if desired, at any position 
on the handle of the rocker so as to enable the motion of the latter 
to be determined. The motion of the upper surface of the ball 
terminating the handle of the rocker may be similarly observed by 
a rolling needle and mirror arrangement. Simultaneous observa- 
tion is also possible of the motion of the end of the handle and of 
the rocker itself, or of any intermediate point on the former. 

III. DESCRIPTION OF THE OBSERVED PHENOMENA. 

Using the method described above, numerous observations 
were made and photographs secured of the mode of vibration of 
the heated rocker under various conditions, and several quanti- 
tative determinations of the frequency and amplitude of vibration 



146 B. N. CHUCKERBUTTI. 

were also obtained. Two different rockers of the usual prismatic 
form, one rather massive and another of medium size, were used, 
and some observations were also made with a light rocker having 
a flat lower surface (instead of a grooved edge). Different lengths 
and diameters of the handle-bar were employed and the effect of 
loading the handle-bar at different points was also investigated. 
Heating the rocker to a greater or less extent, and especially the 
effect of exerting pressure on the surface of the rocker or on the 
handle were studied. Observations were also made using a block 
of rock-salt (which is recommended by Tynclall) instead of lead as 
the support for the heated rocker. It would be unprofitable to 
attempt to describe everything that was noticed in the course of the 
experiments which extended for over four months. The main result 
which emerges from the work is this, that the motion of the rocker, 
especially in the cases in which it gives rise to musical tones, is 
very far indeed from being a simple rocking movement ; the clastic 
vibrations of the rocker and handle-bar play an important part, and 
so far as can be ascertained, entirely replace any such simple rocking 
movement as may occur in the earlier stages of the experiment. 

The photographs reproduced as figs. 2 to 19 in plate will 
give an idea of the* complexity of the vibration curves. These 
were all secured with the massive rocker. The noteworthy feature 
is that the vibrations, when steady, are generally accompanied by 
harmonic overtones which arc of greater or less intensity according 
to circumstances. In the initial stages, when the temperature 
difference is very great, a coarse rocking movement of very large 
amplitude may be observed, but this is ot low frequency and 
hardly gives rise to any audible musical tone. At a slightly later 
stage, the motion becomes of smaller amplitude, accompanied by 
partials evidently due to the elastic vibrations of the system and 
is often rather irregular in character. At a still later stage, the 
vibrations become very regular and are of markedly musical 
character. Figs. 5 to 9 represent the successive stages, the vibra- 
tions being those of the end of the handle bar which was 23 cm. 
long and 6*5 mm. in diameter. Figs. 10 and IT are similar 
photographs with a handle bar 14 cm. long and figs. 12 and 13 
with one of 9 cm. length only, the diameter of the handle bar being 
the same throughout. It is found that the harmonics tend to 
disappear in the last stages of the motion. Figs. 14, 15 an d *6, 



B. N. CHUCKERBUTTI. 



PLATE IX. 



If U 



Fig 5 




Fig 6 



> /! /V A ' 'A / ' 

. y u 



Fig. 7 




Fig 8 




Fig 9 



The successive stages of vibration of the end of the 
handle-bar, 23 cm long and 65mm In diameter. 



B. N. CHUCKERBUTTI. PLATE X. 



Fig 10 







r\ A 

v-v' W Wv/ W VW L-J -, P' 12 



AA/V\AWA/WA r 

WVWWWY 



F,g 14 



Fig 15 



Fig 16 



Fig 17 



Fig 18 



A A- A A A />. 

/ V / Fig 19. 



Vibration Curves of the Trevelyan Rocker, 



Musical Sounds from Heated Metals. 147 

however , give the vibration curves of the block when firmly clamped 
to the rocker itself, the handle bar being the longest of the three. 
It will be seen that the motion of the rocker is, in this case, of 
much smaller amplitude than that of the end of the handle bar 
and is rather irregular in character, and is also accompanied by 
over-tones of high frequency which are not perceptible in the 
motion at the end. Figs. 2, 3, 4 illustrate an interesting effect. 
Fig. 3 shows the motion at the end of the handle, when the rocker 
was vibrating freely. Fig. 2 when the rocker was pressed down by 
a pointed rod so that the pressure on the lead block was greater on 
one of the two edges of the groove. Fig. 4 when the pressure was 
similarly greater on the other edge of the groove. It is seen that 
the phase of the harmonic is reversed in fig. 4. Figs. 17, 18 and 
19 illustrate the motion at the end of the handle bar when the lead 
block was replaced by a block of rock salt. 

The following significant facts were noticed in the course of 
investigation:--- 

1. By simultaneous observation of the motion of the rocker 
and of different points on the handle bar, it is found that the 
vibration curves at different points differ greatly in amplitude and 
character. The motion at the end of the handle is always of greater 
amplitude and indeed sometimes much greater than that of the 
Cocker itself. In the course of vibration, the system divides itself 
into segments. The points of minimum motion are not however 
absolutely at rest but show the overtones very prominently. The 
relation between the movements at different points is not constant 
but changes when the pitch of tone emitted alters. 

2. The musical tone given by the rocker when it is vibrating 
freely docs not change pitch quite gradually but generally passes 
more or less abruptly from one pitch to another, the amplitude ot 
vibration also changing suddenly at the same time. 

3. The effect of pressing the rocker clown on the block is 
always to raise the pitch ; but in some cases, the amplitude of 
oscillation of the rocker actually increases with the rise of pitch, 
and in other cases decreases. When the amplitude increases with 
the rise of pitch as the rocker is pressed down, the loudness of the 
tone markedly increases. 

4. Even a slight pressure on the end of the handle bar which 
rests on the table sometimes results in a marked rise in the pitch 



148 B. N. CHUCKERBUTTI. 

of the tone of the rocker. For instance, in the case of the massive 
rocker with the longest handle, a pressure of only 30 gms. weight 
resulted in an increase of pitch from 140 to 180 vibrations per 
second. 

5. When the rocker vibrates freely, the lower and higher 
limits of pitch of the musical tone which it gives rise to are deter- 
mined by the mass of the rocker as well as by the length of the 
handle bar and its cross section. The shorter or the thicker the 
handle bar, the higher the pitch of the musical sound when it 
begins and the higher when it ends. Diminishing the mass of the 
rocker raises the pitch of the tone. The lower end of the range 
of pitch covered by the musical tones of the freely vibrating rocker 
is generally about the same as or a little less than the pitch of the 
tone given by its elastic vibrations in the gravest mode. The 
higher end of the range of pitch is generally rather ill-defined. 

IV. A CRUCIAI, TEST OF DAVIS'S THEORY. 

The fundamental assumption on which Davis (in the paper 
already cited) bases his investigation is that the motion of the 
healed rocker is a simple rocking movement about the edges under 
the action of its own weight. The observations and photographs 
of the vibration-curves of the rocker discussed above render this 
view untenable and tend to show indeed that it is the elastic 
vibrations of the system that are maintained and that determine 
the pitch of the tone emitted. The measurements of the amplitude 
and frequency of the rocker furnish a crucial test of Davis's theory 
which we now proceed to discuss. 

The motion of the rocker alternately about its two edges under 
its own weight is not an isochronous oscillation. Its amplitude 
decreases (in the absence of a maintaining force) continually on 
account of the loss of energy resulting from the impacts on the 
edges; and the intervals between successive impacts also conti- 
nually decrease. The relation between the period and the ampli- 
tude of the movement of the rocker at any stage may be deter- 
mined from the equation of motion which to a first approxi- 
mation is 



= -Mga 
where M=mass of the rocker, 



Musical Sounds from Heated Metals. 149 

jff^radius of gyration about either of the edges of support, 
and 2a= Distance between the edges. 

Hence, integrating, 

<=rA 



J 



dt 



remembering that, when 

dO 



K*u = ag T l* (I) 

Now, being the greatest angle which the plane containing the 
edges makes with the horizontal surface of the block 

73 

0= , R being the amplitude. 



Hence, since o> increases from o value to u> in the interval T 

t 

2 

Substituting for w in (i) 



w 2ft 

-=-~ or >= 

2 ar 



On the left hand side of this equation, the period of oscillation 
T of the rocker may be determined experimentally by finding the 
pitch of the tone given by it with the sonometer or by recording 
its vibration curves simultaneously with those of a tuning fork. 
On the righthand side, all the quantities arc known except the 
value of R which may be determined from the observation of 
the magnified amplitude of the vibration of the rocker. For if / 
be the movement of the spot of light which we observe and d 
be the distance of the focus from the lens, r the radius of the 
rolling needle, then the displacement of the upper surface of th< 

curved mass of metal upon which the needle rolls is given by ~- . 

2d 

Hence, knowing the distance of this surface from the axis of rota- 
tion of the rocker and the distance between the two ridges, we 
may deduce the magnitude of the vertical displacement of the free 
edge from the surface of the block. Thus it is possible to determine 
directly the quantities on either side of equation (2) and to test its 



150 B. N. CHUCKERBUTTI. 

validity and therefore also the validity of Davis* s theory. In the 
table below, the first column gives the serial number of observation 
(the handle of the rocker being of different length in some of these 
cases). The second column gives the frequency of the tones as 
determined by the sonometer. The third gives the value of the 
magnified motion (i), the fourth column gives the displacement of 
the upper surface of the curved mass of metal, the fifth column gives 
the vertical movement 7? of the edge of the rocker, and in the 
sixth column we have the value of the frequency calculated from the 
equation (2). It will be noticed that the observed and calculated 
frequencies differ very widely, and it is clear, therefore, that the 
irequcncy of the oscillation is not determined by the motion of 
the rocker under the action of its weight. To exhibit this even 
more clearly, column 7 of the table shows the value for /, the 
movement of the spot of light, that would be necessary to give the 
observed frequency of oscillation if equation (2) were to hold good. 
It will be found on comparison of column (3) and (7) that the 
observed oscillation is of far smaller magnitude than deduced from 
the observed frequency. 

DIMENSIONS OF THE ROCKER. 

2a Distance between the ridges 

=0*6 cm. 
Hase of the triangular section 

=5 cm - 
Length of each side of the triangle 

=3-4 cm. 

Height of the C.G. above a line joining the ridges 
===6=i'4i cm. 

/f-3'47 

(Radius of the needle 7=0-042 cm. 
d==gS cm.) 



Musical Sounds from Heated Metals. 



I 


2 


3 


4 


^ 


o 


7 


No. of 
Observa- 
tions. 


Frequency 
by 
sonometer. 


I 

(in cm.). 


Displace- 
ment 
rl 
7 cm. 
2d 


Vert, dis- 
placement 
of the ridge, 
(in cm.). 


Frequency 
(calculated). 


Value for I 
that would 
fit in equa- 
tion (2). 


, 


US 


075 


0*00016 


0-000018 


30i 


5-05 cm. 


2 


121 


070 


CTOOOI5 


0-000017 


3ii 


4-7 cm. 


3 


130 


o 64 


0-00013 


0-000015 


33i 


4 - o cm. 


4 


136 


o 60 


O "COO 1 2 


0-000013 


342 


3-62 cm. 


5 


145 


0-50 


O'OOOIO 


O'OOOOII 


3i 


3-20 cm. 


() 


162 


0-40 


o 00008 1 cvooooocj 


420 


2 -6 cm. 


? 


l8o 


O"2O 


0-00004 


0-000005 j 595 


2 cm. 



The foregoing conclusions are not invalidated on taking into 
account the maintenance of the motion by the periodic expansions 
and contractions of the block of lead on which the rocker rests. 
For, to a first approximation, the period of the maintained oscilla- 
tion must be the same as that of the unmaintained motion having 
the same amplitude. It can readily be shown that according to 
Davis's theory, the actual expansions and contractions of the lead 
block necessary to maintain the oscillation are small compared 
with the motion of the free edges of the rocker. This may be done 
in the following way : 

Let > and w' be the angular velocities just before and just 
after impact of the rocker upon one point of the lead block (say 
0). Then the relation connecting them is 1 

K, a + b l + a* , 



KI being the radius of gyration of the rocker about the centre 
of gravity. If the vibrations are continuous, the angular velocity 
acquired just before the next impact upon the other point of 
support (say 0') must be equal to o>. Immediately after the impact 
upon O, the point begins to rise. Suppose, it has risen through a 



Davis, loc. cit. 



152 13. N. CHUCKERBUTTI. 

height A in time t. Then it can be shown that the equation of 
motion of the rocker is 



that is *S=- a ( g + S) 

where K,=radius of gyration about an axis passing through the 
centre of gravity, 

and A=radius of gyration about one of the edges of the rocker. 
Hence integrating and noticing that 

do 

when / = o, =o> 

dt 

we have 7< a agt a + /v'V 

Integrating again and noticing that 
when t o, A = o, and 

(H being the height through which rises), 

, ,,,. a?t l , . ,,-> ,. jr^H 

we h.ivt 4 A 1 B = an -f K L ^ t + A . 

2 2tf 

Now, just before the next impact will have risen to a height 
//, and ()' will have sunk down to its original position. Therefore 

the value of and -~ will be - and - < respectively. The time 
dt 2 a 

for which the rocker is in contact with the block being 772 where 
T= period of vibration, the above relation takes the form, 

K* H = C ^ + all -!<><>>'*/, 



whence 



or 



where. 



Musical Sounds from Heated Meta!s. 153 

Substituting in this equation, the value for u> as obtained before, 
viz. 



From equations (2) & (3) we have 

7? K*-a* 3-47 --OQ 

_ = __ __ J \/ _ 2. r=r O 7 

H 2K\I-C) 3-47 x 'I J '' 

The free motion of the edges of the rocker required on Davis'-; 
theory is thus about ten times the expansion of the supporting 
block necessaiy to maintain it, and the influence^ the latter on 
the period of the motion under gravity is therefore quite negligible. 
Thus we finally arrive at the conclusion that the observed 
amplitudes and frequencies are not connected together in the 
manner necessary for the validity of Davis 's theory, and the 
discussion supports the view that the frequency of oscillation of 
the rocker is determined principally by its elastic vibrations. 

The two facts now remaining to be explained are (i) the rise 
of the pitch of the tone of the rocker as the motion dies away and 
(2) also, the rise of pitch resulting from exerting pressure on the 
rocker. It does not seem difficult to reconcile these with the view 
that the pitch is determined by the elastic vibrations of the system. 
For, the contact of the rocker with the lead block at one end and of 
the handle with the table at the other end no doubt operates as a 
constraint tending to raise the pitch of the elastic vibrations to a 
greater or less degree depending on the effectiveness of the con- 
straint. Exerting a pressure on the rocker by the point of a rod 
resting upon it would similarly operate as a constraint raising the 
pitch of the vibration to quite a considerable extent. 

V. SUMMARY AND CONCLUSION. 

The paper describes an experimental study and a theoretical 
discussion of the vibrations of the Trevelyan Rocker, and the 
manner in which the musical tones given by the rocker arise. A 
method is described by which the vibration curves at different 
parts of the rocker may be observed and photographed and 
numerous illustrations are reproduced. The theory of the rocker 
worked out mathematically by Davis is critically discussed, and 



154 B- N. CHUCKERBUTTI. 

it is shown that the basis on which it rests, namely, that the motion 
of the heated bar is a simple rocking about the two edges under 
the action of gravity is contradicted by observations on the 
amplitude and frequency of the vibration of the rocker. The 
outcome of the research is to show that the musical tones princi- 
pally arise from the clastic vibrations of the system composed of 
the rocker and its handle, and its pitch is determined by these vibra- 
tions. As the vibrations occur under constraint, the pitch may 
vary to some extent with the experimental conditions. The 
effect of pressure on the rocker in raising the pitch is similarly 
explained. 

In conclusion the author wishes to record his best thanks to 
Prof. C. V. Raman who suggested the investigation and kindly 
provided all facilities for work at the University College of Science. 

Dated Calcutta, i 
The i8th October, 1920. ) 



XIL Some New Illustrations of Optical 
Theory by Ripple Motion. 



By Rajendra Nath Ghosh, M.Sc. 



(Plate XI.) 



CONTENTS. 
I. Introduction. 

II. Description of the New Ripple Apparatus. 

III. Interferences near Caustics and the Theory of the Rainbow. 
IV. Anamolous Propagation near Foci. 

V. Laminar Diffraction. 

VI. Summary and Conclusion. 

I. INTRODUCTION. 

Since Vincent published his well-known photographs of 
ripples presenting analogies with various optical phenomena, the 
subject has' continued to attract attention from other workers who 
have used the same or different methods. A fairly comprehensive 
bibliography of the literature will be found in a recent paper by 
Watson and Shcwhart. 1 In the present paper will be described 
some further studies in the same field carried out by the author 
while working at the Indian Association for the Cultivation of 
Science, Calcutta. The research was undertaken with the aid of a 
new apparatus which is very compact and serves as a convenient 
Jiethod of producing ripples and observing them by intermittent 
illumination, the frequency and amplitude of the ripples being both 
capable of being rapidly altered without interference with tho 
stroboscopic arrangement. The design of the apparatus is due to 

' Physical Review, February 1916, pa^e 226. 



156 R. N. GHOSH. 

Prof. C. V. Raman and is somewhat similar to that described by 
him in a recent paper on the experimental study of vibrations. 1 

II. DESCRIPTION OF THE NEW RIPPLE APPARATUS. 
lug. i in Plate XI is a picture of the apparatus used for 
exciting ripples, part of the ripple tank also appearing in the 
photograph reproduced. An electric motor causes a horizontal 
pivoted lod to oscillate rapidly in a vertical plane, a dipper hanging 
from the end of this rod exciting the ripples on the surface of the 
liquid contained in the tank seen to the right in the picture. The 
motor also carries a card-board disc with a sector cut out in it 
through which the full beam of an electric lantern can pass once in 
each revolution of the motor. The beam of light is reflected by 
means of mirrors and comes up through the glass bottom of the 
tank, the liquid used in it being generally water. The ripples are 
then seen on a screen held vertically above the tank. They may, 
if desired, also be projected horizontally on an ordinary lantern 
screen by using a mirror held at 45 above the surface of the tank 
to relied the light at right angles to its path. Since the Hashes 
of light aie synchronous with the oscillations of the dipper, the 
ripples appear perfectly stationary. The frequency and wave- 
length of the ripples may be altered without interfering with the 
running of the apparatus by merely increasing or decreasing the 
speed of the motor by adjustment of a rheostat kept in circuit 
with it. The amplitude of the ripples excited may be adjusted to 
any desired value and kept constant by moving the crank-pin on 
the shaft of the motor to the position necessary and then fixing it. 
As the arrangement is very robust, any desired form of clipper 
may be put on without prejudice to the working of the apparatus. 
Since the full beam of the lantern can pass through the aperture 
in the stroboscopic disc, there is an abundance of light for all 
purposes. The device for altering the wave-length of the ripples 
is very useful in illustrating the theory of interference and diffrac- 
tion phenomena on the screen by lantern projection. The ripples 
are seen perfectly at rest and can be photographed from the screen 
with any ordinary camera. 



Physical Review, Nov. 1919. See also these Proceedings, Vol. VI, Plate V. 



N. GHOSH. 



PLATE XI. 





Fig. 3 





Fig. 4 



Ripple Apparatus and Photographs. 



Optical Theory and Ripple Motion. 157 

III. INTERFERENCES NEAR CAUSTICS AND THE THEORY 
OF THE RAINBOW. 

The geometrical theory of the rainbow given by Descartes 
accounted for the principal bow qualitatively. The more complete 
theory (involving interferences) was given by Young who thus 
explained the supernumerary bows. Airy showed that the wave- 
front emergent from the drop has the form of an inflected curve, 
and according to his calculations, the direction of the principal 
maximum of intensity is slightly different from the normal to the 
inflectional tangent, which is the direction (accordingly to the 
geometrical theory) in which the least deviated ray travels. The 
supernumerary bows are the interference fringes formed alongside 
the caustic, and they decrease gradually in width and intensity as 
we move away from the principal fringe. 

The theory of the rainbow may be beautifully illustrated by 
means of ripples. For this purpose, the dipper used to excite the 
ripples is given the form of the inflected wave-front emergent from 
the drop, of course on a much larger scale. The dipper should be 
carefully shaped by drawing the desired form of carve and bend- 
ing a strip of brass or copper to its shape, and it should just touch 
the surface of the liquid. The ripple as it starts out from the 
dipper lias the form of an inflected curve, but in proceeding out- 
wards it doubles up and forms two superposed ripple-trains which 
interfere, giving maxima and minima of gradually decreasing 
width and intensity. This effect is clearly shown in fig. 2 in 
Plate XI. The photograph clearly shows how the ripples from the 
concave half of the strip converge to a train of foci, then diverge 
again, and ultimately become superimposed on the ripples diverg- 
ing from the convex half, giving interferences. The great width 
and clearness of the principal fringe, and the weaker and narrower 
fringes lying on one side of it are clearly shown. In agreement 
with theory, the minima are practically lines of zero amplitude, 
and the direction of the maximum disturbance is also deviated a 
little away from the normal to the curve at the point of inflection 
and towards the first minimum lying on one side of it. An interest- 
ing feature will also be noticed in the photographs that the ripples 
in the two regions of maximum lying on either side of each 
minimum are displaced relatively to each other. This occurs to n 



158 



R. N. GHOSH 



greater or less extent in all cases of interference, and is closely 
connected with the question of energy flow in the field. 

The intensity at any point in the interference field due to an 
inflected wave-front can be readily calculated by elementary 
methods without carrying out the usual elaborate integration over 
the wave-front first worked out by Airy. 

Let Oy be the normal to the inflectional tangent xOx', the 
equation of the wave-front being y==Ax*. Let P and P' be corres- 
ponding points at which the wave-normals are parallel to each 
other and inclined at an angle to the y-axis. We may assume 
without appreciable error that the effects of different parts of the 









wave-front are propagated as rays in the direction of the different 
wave-normals in accordance with the laws of geometrical optics. 
The curvatures of the wave-front at P and P' are equal and of 
opposite sign, and hence at a sufficiently great distance the effects 
contributed in the direction of the wave-normals at these points will 
be of equal amplitude. These effects arc proportional to a { where 



On taking the square root, the value a { corresponding to the nega- 
tive sign becomes imaginary. We may interpret this as a gain of 
phase of ?r/2 which occurs in the rays from the part of the wave- 



Optical Theory and Ripple Motion. 159 

front which changes the sign of its curvature in passing through a 
locus. The numerical value of a Y is 



The path difference of the rays proceeding from P and P' can 
then be easily shown to be 



The amplitude of the resultant of the two will be 



On squaring this, we get the intensity in the direction 0, and it can 
easily be shown that equation (2) gives results which, for positive 




M 

FIG. 6. 

values of greater than a certain very small limit, are for all 
practical purposes the same as those given by Airy's integral. In 
fact the result given by (2) is identical with that obtained from 
Airy's integral by the method of semi-convergent expansions. 

The most convenient experimental arrangement for observing 
on the optical bench the different steps in the transformation of 
an inflected wave-front to a cusped wave showing the interferences 
alongside the caustic is that indicated diagrammatically in fig. 6. 

5 is the slit which is the source from which light diverges and 
falls upon the reflecting cylinder C. M is a short focus lens which is 
placed so that its focal plane is well in advance of the edge of the 
cylinder C grazed by the incident rays. The rays after reflection 
at the surface of the cylinder form a virtual caustic within the 
cylinder. After passing through the lens they form an inflected 
wave-front which in the region behind, forms a cuspecl wave-front 



i6o 



R. N. GHOSH. 



and shows the interferences alongside the caustic very well. 1 By 
interposing a wire of greater or less diameter immediately behind 
the lens, different portions of the inflected wave-front can be cut 
off. As the wire is moved across the field from one side to the 
other, its shadow moves in the interference-pattern towards the 
caustic and then retraces its path y the fringes practically disappear- 
ing from the region covered by the shadow/ 1 showing thereby 
clearly that the inflected wave doubles up and forms a cusped 
wave-front in the course of its propagation and thus gives rise to 
interferences. 

The formula (2) given above for the disturbance in the inter- 
ference-field due to an inflected wave-front was tested experiment- 
ally with the ripple apparatus by measuring the angular positions 
of the minima of disturbance relatively to the inflectional tangent, 
and also calculating the same from the observed wave-length of 
the ripples and the known form of the dipper. The results are 
shown in the table given below : 

TABLE I. A=I '070111. A=ar^. 



Minimum No. 


i 


2 , ^ 


4 


5 






, 





._ 


Pontoon observed 


66 


50 33 


29 


21 


,, calculated 


68 


52 40 


28 


23 



The agreement is fairly satisfactory. 

IV. ANAMOLOUS PROPAGATION NEAR Foci. 

Gouy discovered in 1890 3 that a spherical wave gains in phase 
by half a wave-length in passing through a focus. His experiments 
were repeated by Fabry* and Zeeman. Later Sagnac 5 and Reiche 
again performed the experiment. Sagnac on the basis of diffrac- 



1 See N. Basu, Phil. Mag., Jan. 1918, page Q2. 

2 The fringes elsewhere are also disturbed by the diffraction effects due to the 
wire, but not to such an extent as to obscure the general nature of the case. Some 
interesting effects are observed when the wire just coincides with the point of 
inflection in the wave-front. 

o Gouy, Ann de Chitn et de Phys., 6th series 24, pp. 145-213. 

* Fabry, Journal de Phys., 3rd series 2, p. 22 (1892). 

* Sagnac, Journal de Physique, Oct. 1903. 



Optical Theory and Ripple Motion. 161 

lion theor}' explained the gain of phase as due to the oscillation 
of phase and intensity ; Reiche ! on the electromagnetic theory 
showed that the electric vector in passing through the focus changes 
sign while the magnetic vector remains unchanged, indicating a 
gain of phase of half a wave length in the passage through the 
focus. Since it is now established that a wave gains in phase in 
passing through a focus, it appeared worth while to see whether it 
is possible to detect the phenomenon in the case of ripples as well. 
With this object, the dipper was made into the shape of an arc of 
a circle of radius 3*5 cm. When a plane wave passes through a 
lens, it is transformed into a concave wave which first converges 
to a focus, and then diverges into the surrounding medium in the 
form of convex waves. The waves generated by the dipper behave 
sitnilaily. Fig. 3 in the plate shows the effect. Careful observa- 
tion showb that the wave lengths near the focus are slightly longer 
than the others. This can be interpreted as equivalent to an 
acceleration of velocity near the focus, so that the waves gain a 
certain fraction of wave-length in phase in passing through it. 
Using very small amplitudes of excitation of the ripples so as to 
avoid errors due to the alteration of the velocity of the ripples 
which occur at larger amplitudes, a large number of measurements 
were made with the ripple apparatus, the total space occupied by 
three waves at the focus, and three waves on either side of it being 
determined. The former was found to be greater than the mean of 
the two latter by about quarter of a wave-length, which is what 
we should expect in the case of cylindrical waves in the optical 
case. It may be remarked that this gain in phase in passing 
through a focus is also indicated by the good agreement between 
theory and experiment in the case of the rainbow (dealt with in 
the preceding section) where the phase-change was taken into 
account in finding the positions of the minima of illumination. 

It may be remarked in passing that fig. 3 also clearly shows 
the diffraction-effects in the neighbourhood of the focus. It is 
precisely because of diffraction that it is necessary to take into 
account a sufficient number of waves on either side of the focus in 
order to determine the gain in phase. 

1 Reiche, Ann. d Physik, April and June 1909. (An account is given in Wood's 
Optics, page 263.) 



162 R. N. GHOSH. 

V. LAMINAR DIFFRACTION. 

In some recent studies of laminar diffraction phenomena, 1 it 
has been found that the edges of thin laminae often diffract light 
in a strongly unsymmetrical manner., a great deal more light being 
scattered towards the retarded side of the wave-front than on the 
other. It was thought that it would be of interest to try and 
illustrate this phenomenon with the aid of the ripple apparatus. 
A dipper was used of the form of a long i so as to 

imitate the discontinuous form of wave which presumably arises 
when a plane wave passes through a semi-infinite transparent 
lamina with a sharp straight edge. The result obtained is illus- 
trated in fig. 4 of the Plate. It will be seen that the disturbance 
arising from the region of bend in the dipper is overwhelmingly 
large to the region on the right which corresponds to the retarded 
side of the wave-front, while on the left of the normal drawn to the 
line of the dipper, the ripples are almost perfectly straight with 
hardly any sensible disturbance. Reversing the position of the 
dipper so that the retarded side of the ripple was on the left 
immediately resulted in a corresponding reversal of the pattern 
showing that the effect was genuine. 

VI. SUMMARY AND CONCLUSION.* 

The paper describes the construction and use of a very simple 
and compact apparatus which enables ripples to be excited on the 
surface of a liquid and observed stroboscopically, the frequency 
of the ripples being capable of rapid alteration while the apparatus 

' P. N. Ghosh, Proc. Roy. Soc. London, A scries, Vol. 96, 1919, page 261 ; 
Proc. Indian Assoc. for the Cultivation of Science, Vol. VI, part I, page 52. 

2 It may be of interest to note here that since the return of Mr. R. N. Ghosh to 
Allahobad to join his appointment at the Ewing Christian College, the work with 
the ripple apparatus used by him has been continued at this laboratory. 
Mr. Goverdhaii Lai Datta has shown that the apparatus can be used to exhibit the 
analogy of the effect of groove form on grating spectra, and has succeeded in 
constructing a ripple grating which concentrates nearly all the energy in a single 
spectrum. Mr. Datta has also studied the character of interference and diffraction 
phenomena in the immediate neighbourhood of the sources which had not previously 
received adequate attention. Some very interesting results have also been obtained 
by Mr. J. C. Kameswara Rao on the forms of ripples of large amplitude. These 
are described in a paper appearing later in this volume. The work done has 
demonstrated the convenience and utility of the apparatus in the exact study of 
ripple motion. (C. V. R., 26 Oct., 1920.) 



Optical Theory and Ripple Motion. 163 

is running without interfering with the stroboscopic arrangement. 
The amplitude of the ripples excited can also be varied and kept 
strictly constant. 

With the help of the apparatus the following new illustrations 
of optical theory by ripple motion have been worked out: (i) 
Interferences near Caustics and the theory of the Rainbow; (2) 
Anamolous Propagation near Foci ; (3) Unsymmetrical Types of 
I y aminar Diffraction, and photographs of these cases are reproduced 
with the paper. 

In conclusion, the author wishes to record his grateful acknow- 
ledgments to Prof. C. V. Raman for his suggestions and constant 
encouragement during the progress of this research. 

DATED CALCUTTA, > 
. ) 



The i$th of ]unc y 1020. 



XIII. The Theory of Impact on Elastic Plates. 



By K. Seshagiri Rao, B.A. (Rons.), Research Scholar in the 
Indian Association for the Cultivation of Science, Calcutta 



SYNOPSIS. 

1. Hertz's Theory: In a paper contributed to the Physical 
Review for April 1920, C. V. Raman has shown how Hertz's theory 
of the collision of elastic solids may be extended to the case of 
transverse impact on elastic plates so as to determine the propor- 
tion of the kinetic energy of the impinging body transformed to 
energy of wave motion in the plate. The assumptions forming 
the basis of this extension of Hertz's theory have been tested 
experimentally by the present author and it is shown how the 
residual discrepancies between observation and the results given 
by Raman's theory in the case of relatively thin plates may be 
explained. 

2. Duration of Impact : Except in the case of relatively thin 
plates the duration of impact is found to be substantially the same 
as that given by Hertz's theory of impact on an infinite mass of 
solid with a plane face; however, when the thickness of the plate 
is diminished to nearly the critical value at which the whole of 
the energy of the impinging body is transformed into energy of 
wave-motion, experiments show an appreciable increase in the 
duration of contact. It is shown that this is what might reason- 
ably be expected on theoretical grounds. 

3. Apparent Coefficient of Restitution: Allowing for a small 
proportion of energy dissipated otherwise than in the production of 
wave motion in plate, and also making a correction for the increase 
of duration of impact referred to above, good agreement is found 
between theory and experiment for the values for the apparent 
coefficient of restitution. The increase of the coefficient of resti- 



166 K. S. RAO. 

tution for decreasing velocity of impact is quantitatively ex- 
plained. 

4. Acoustic Applications: The theory with suitable modi- 
fications appears to be capable of being applied to the problem of 
the analysis of the vibrations of bells or plates excited by impact. 

I. INTRODUCTION. 

The mathematical theory of the collision of elastic solids 
given by Hertz is primarily applicable only to cases in which the 
shape of the impinging bodies and their velocity of collision is such 
that a negligible proportion of their kinetic energy is transformed 
into energy of wave motion in the solid or dissipated in producing 
quasi-permanent deformations. It is naturally of interest to deter- 
mine how Hertz's theory has to be modified in various cases where 
such transformations of energy do occur to an appreciable extent. 
In a paper recently contributed to the Physical Review for April 
1920, Prof. C. V. Raman has shown how the problem of trans- 
verse impact of a solid with curved faces on a plane elastic plate 
may be dealt with by an extension of the method suggested by 
Hertz, and has given a formula for the apparent coefficient of 
restitution of the impinging solid from which the proportion of its 
kinetic energy transformed into energy of wave motion in the plate 
may be calculated. In this application, it was assumed that the 
duration of impact was to a first approximation the same as that 
given by Hertz's theory for impact 011 the plane face of an infinite 
mass of solid. From the theory the result w;is deduced that as the 
thickness of the plate is gradually diminished, the apparent coeffi- 
cient of restitution should continually dimmish till when the thick- 
ness is below a certain critical limit depending upon the elastic 
constants and upon the mass, dimensions and velocity of the im- 
pinging body, the coefficient of restitution should vanish altogether : 
in other words that the body should then behave as if it were per- 
fectly inelastic and remain in contact with the plate. It is obvious 
that when this critical limit is reached the duration of impact 
should become infinite : in other words that the very assumption on 
which theory itself is based should break down at or near this point. 
As a matter of fact, observation shows that while the calculated 
coefficient of restitution is in excellent agreement with the theory 
for plates of moderate thickness, deviations appear when the thick- 



The Theory of Impact on Elastic Plates. 167 

ness is rather near the critical limit and in such cases the observed 
coefficient of restitution falls less rapidly to zero with decreasing 
thickness of the plate than the calculated coefficient. It would 
appear from these considerations that in the case of the relatively 
thin plates the duration of impact should be somewhat larger than 
that given by Hertz's theory, and that this should be taken into 
account in the calculation of the apparent coefficient of restitution 
and that a better agreement might then be expected between 
theory and experiment in the case of such plates. The present 
paper describes work undertaken by the author mainly to clear up 
the outstanding discrepancies between theory and observation on 
the lines explained above. 

II. SOMK EXPERIMENTAL REsui/rs. 

The materials selected for the investigation consisted of a set 
of five mild steel plates of thicknesses 3*82, 2-30, r8r, i 21 and 
0*60 cms. respectively. The impinging body was a sphere of 
hard steel of diameter 2*70 cms. 

The first set of experiments consisted in a determination ot 
the duration of impact between sphere and plate for different 
velocities of impact. This was done in the usual way by allowing 
an electric circuit to be completed by the contact between the 
plate and the ball, the total quantity of electricity which passes 
from the time they meet till they again separate being measured 
on a ballistic galvanometer. Without taking into consideration 
the corrections for self-induction we get the formula 

n V RQ 

V='R T '=TT 

where R is the resistance, Q quantity of electricity and T the time 
of impact. The correction for self-induction arc two in number. 

(a) The current instead of rising instantaneously to its full 
value when the ball and the plate meet grows exponentially. 

(b) When the ball separates from the plate a spark may be 
produced so that the current does not fall instantaneously to zero. 

These corrections can be determined as follows : If a fairly 
large resistance be included in the circuit we can consider the 
contact resistance as negligible compared to the first. It follows 
therefore when the ball impinges the resistance rises to its final 



1 68 



K. S. RAO. 



value almost instantaneously. We may therefore treat the circuit 
as one having constant resistance. The equation of growth will 
then be the usual expression 



L di 

L df 



V = o 



or 



R 



Integrating 



R 



The correction is thus a constant addition to the time. The 
method of investigating these corrections wns the same as used by 
Scars ! for the problem of longitudinal impact of rods. The com- 
plete connections arc shown in the figure. 




FIG. T. 

13 was a battery of 4 volts and R was a resistance box from 
which 5000 ohms were unplugged. C was the condenser used in 
calibrating the galvanometer. With the high resistance used it 
was found that there was no sparking effect at all while the correc- 
tion for self-induction was very small. 



1 Proceedings of Cambridge Philosophical Society, Vol. XIV, page 273. 



The Theory of Impact on Elastic Plates. 160 

The ball was suspended by a fine covered copper wire which 
was soldered to it, and which also served to convey the current 
during the impact. The distance of impact was regulated by 
holding the ball in position by an electro-magnet, care being taken 
that there was no leakage through the electro-magnet or through 
the suspension wire. Both the sphere and the surfaces of the 
plates on which the sphere impinged were highly polished and 
cleaned. It was found that concordant results could be obtained 
only when these were scrupulously clean, the slightest trace of 
dirt producing great variations in the readings. 

For every plate the duration of impact was determined with 
different velocities, In every case it is found that the duration 
varies as the fifth root of velocity in the manner required by 
Hertz* s theory. It is also found that the duration of contact is 
practically the same for the plate^ having thickness from 3*80 to 
1*21 (though a very slight increase is to be noticed for decreasing 
thickness) ; but suddenly increases by io% for the plate having a 
thickness of O'6 cms. The table showing the results is given 
below. 

TABLE I. 

The height of suspension of the ball 74*35 cms. 
Voltage applied 4-23. 

Plate No. i, Thickness 3*82 cm. 



Withdrawal Galv. Throw 



97-0 divisions. 

i 

4*2 

I 

2? ,, i 129-0 ,, i i5 8 5 

136-0 ,, 1 58' i 



108-0 ,, 157-2 

i 



170 



K. S. RAO. 

Plate No. 2, Thickness 2*30 cms. 



Withdrawal 

a 

7-3 cm. 

5'9 -. 

2'7 ,, 



Galv. Throw 
e 


l/T.e 


105 


156-3 


no 


157-0 


I IQ 


157-5 


130 


158-0 


146 


158-0 



Mean 157-6 



Withdrawal 



7 '8 cm. 



Plate No. 3, Thickness i'8i cm. 

Galv. Throw | 

e : 

i 

104 ! 

109 5 
114 

122 



Va . 6 



159-0 



1577 



Mean 158-2 



Plate No. 4, Thickness 1*21 cms. 

Withdrawal I Galv. Throw 



t/7. 



7-3 cm. 


106 o 


158-8 


V9 - 


119-0 


164-8 


3* > 


129 


i6i'4 


2'2 ,, 


135 


157-8 


3'3 ., 


128 


1 62 '4 



Mean 160-6 



The Theory of Impact on Elastic Plates. 
Plate No. 5, Thickness 0*60 cm. 



171 



r ithdrawal 


Galv. Throw 






A 


V a . 8 


a 


8 




7-6 cm. 


118 


177-0 








6-3 ,, 


121 


174-0 


5- ,. 


120 


174-0 


V7 ,, 


134 


I 75 H 


27 ,, 


142 


i i 74' 5 


i-5 


162 


175*4 



Mean 175-1 

It will be found that the readings in last column are p racti- 
cally constant for each plate and its change is little except for the 
thinnest plate. 

If we take the mass of the plate to be very great and calculate 
the duration of contact on the basis of Hertz's theory, we find the 
cal culated value agrees very well with the thicker plates while 
for the thinnest the discrepancy is about 10%. 

TABUS II. 

Velocity corresponding to 6 cms. withdrawal 21*78 cms -/sec. 

I 



Calculated from 
Hertz's theory. 



1-265 x 10- * 



OBSERVED VALUES OF DURATION. 



Plate No. i. 



io-* 



No. 2. 



1-256 x io- 



No 3 



P25Q X 10-* 



No. 4. 



1-281 x io 



No. 5. 



i 190 x io ~ 



The second set of experiments consisted in the determination 
of the coefficient of restitution for different velocities of impact. 
On Raman's theory the expression for e the coefficient of restitu- 
tion is given by 



4- kM 



172 



K. S. RAO. 



where k a constant depending on the form of the transverse wave 
in the plate and 



where 



/:= Young's modulus, 

r^ duration impact, 

" = Poison's ratio, 

2/ = Thickness of plate, 

/> = density, 
M = the mass of the impinging sphere. 

It will be evident from the expression that e varies with T, i.e- 
with the velocity of impact, e was determined experimentally for 
each plate for different impinging velocities and was calculated from 
the above expression, k being taken to be 0-56. T, the duration of 
impact, is taken in the calculation for the first four plates to have 
given the value by Hertz's theory. For the thinnest plate, the 
calculated coefficient of restitution depends on whether T is to be 
taken the actual value as found in experiment, or the value as 
given by Hertz's theory. Both the values are given, the former 
being marked by asterisks. 

TABUS III. 



Vel. 2T4 Vel. 


21-8 


Vel. 


18-2 


Vel. 14-5 


Vel. iO'9 


Plate 














thickness 


\ ' 




___ 






- 


_ -__ 


I Calc. | Obs. Calc. 


Obs. 


Calc. 


Obs. 


Calc. 


Obs. 


Calc. 


Obs. 




1 


















i 
















3-82 cin. . . oc 


>7 0-91 0-97 


0-91 


0-97 


0-91 


098 


0*91 


0-98 


0-92 


2-30 ,, i o-c 


>3 0-87 0*04 


o 88 


0-94 


o-8K 


0-94 0-89 


0-94 


0-89 


1-81 , t cr* 


<8 0-83 , 0-88 


0-83 


0-89 


, 0-84 


0*89 0-84 


0-90 


0-84 


1-20 ,, .. o-; 


7 070 077 


071 


078 


072 


079 072 


079 


073 


0-60 ,, . o-i 


I* 0*30 0*32* 


0-31 


o'33* 


0-32 


0-35* 


0'34 


0-37* 


0-30 


0'' 


54 0-36 




0-38 




0'39 




0-4.2 





In considering the degree of agreement of the calculated and 
observed coefficients of restitution shown in Table III it must be 
remembered that Raman' s formula takes account of only the trans 
formation of kinetic energy of translation into the energy of wave 



The Theory of Impact on Elastic Plates. 173 

motion in the plate and does not take into consideration the energy 
dissipated in production of permanent deformation in the plate. 
Even for the very moderate velocities of impact dealt with in the 
present investigation, some dissipation of the latter kind is evitable, 
distinct marks being left on the mild steel plates on the result o[ im- 
pact, and we may reasonably expect the observed coefficient of res- 
titution for the thick as well as for the thin plates to be less than the 
calculated value by some units in the second place of decimals. This 
is exactly what appears in table III, in the case of the four moder- 
ately thick plates, and also for the thinnest plate if we take the 
asterisked values of the calculated coefficient for comparison with 
the experimental values. 

DATED CALCUTTA, 
The i^th of May, 1920. 



XIV. On Ripples of Finite Amplitude. 



By J. C. Kamesvara Rav, M.Sc., Palit Research Scholar 
in the Calcutta University. 



(Plate XII.) 

CONTENTS. 

SECTION I. Introduction. 
SECTION II. Description of Apparatus. 

SECTION III. Observation of the Form of Ripples of Large Amplitude. 
SECTION IV. Mathematical Theory for Plane Waves of Finite Amplitude. 
SECTION V. Mathematical Theory for Diverging Waves of Finite Ampli 

tude. 
SECTION VI. Dependence of Velocity of Ripples on Amplitude and 

Depth of I/iquid. 

SECTION VII. Outline of Further Research 
SECTION VIII. Synopsis. 

SECTION I. INTRODUCTION.* 

The problem of wave motion on the surface of liquids has for 
a long time been the subject of both theoretical and experimental 
study, its practical interest as well as the facility with which the 
various phenomena connected with it can be observed contributing 
to the development of the subject. Scott Russel 1 was one of the 
earlier observers in this field. Soon after Stokes* gave a complete 
theory of the oscillatory waves of finite amplitude and of per- 
manent type. He found that to a second approximation, the 
velocity of waves does not depend upon the amplitude of the 

* Note. The experimental work described in this paper was carried out in June 
and July 1920. The mathematical theory was presented at a meeting of the Cal- 
cutta Mathematical Society on the 4th of September, 1920. The effects have since 
been observed also with mercury surfaces. 

Brit. Ass. Rep., Vol. VI, 1844. 

Carnb. Trans, t. VIII (1847) and papers, Vol. I, p. 197. 



176 J. C KAMESVARA RAV. 

waves, and as regards form he found 1 that the hollows were no 
longer similar to the crests, but the height of the latter exceeds 
the depth of the former and that the crests are narrower than the 
hollows. For a third approximation he found that the velocity of 
propagation increases with the amplitude. In his investigations, 
he did not however take into account the effect of capillarity, but 
Kelvin l solved the problem completely for small amplitudes and 
proved the existence of a minimum velocity of propagation. 
Later on Raylcigh* studied ripples experimentally and observed 
them under intermittant illumination by the method of Foucault's 
test. By this method he measured the surface tensions of some 
liquids and the effect of contamination 011 the surface tension of 
water. Vincent 8 studied ripples experimentally and took many 
beautiful photographs, illustrating interference, diffraction, etc., 
and more recently Watson* used an improved method of Rayleigh's 
for the determination of surface tension and viscosity of liquids. 
Dr. Wilton 6 gave a theory of the form of ripples of finite amplitude 
and short wave lengths and he .showed mathematically that the 
length of the troughs should be very small compared with the 
length of the crests, and in fact the form of the troughs according 
to him have the shape of a cusp (ceratoid), where the curvature is 
very great. In the case of fairly large wave lengths (say 2-5 cm.) 
he obtained two forms theoretically, one similar to those of shorter 
ripples and the other with two crests in each wave length, and 
suggested (wrongly as it would appear) that the latter form was 
unstable. 

It was with a view to test the accuracy of Wilton's work that 
the present investigation was first taken up, but in the course of 
the research, some quite new and interesting results were obtained. 
It was found that the form of ripples excited by a simple periodic 
pressure at a point were not simple undulations but with the 
increase of amplitude and wave length they assumed very complex 
forms varying with the conditions. The paper describes the effects 
observed and the mathematical theory is also worked out. 



l Baltimore Lectures, p. 598 ; Phil. Mag (4) t. XLVII, p. 374. 
a Phil. Mag. ($) t. XXX, p. 386; Papers t. iii, p. 394. 
3 Phil. Mag. Vol. 43, p. 417 ; 45, 191 , 46, 290. 

* Phy. Rev. Vol 12, p. 257 (1901); Piiy. Rev. Vol. 15, p. 20 (1902); Phy. Rev. 
2nd series, Vol. VII, p. 226 (1916). * Phil. Mag. May 1915, p. 688. 



On Ripples of Finite Amplitude. 177 

SECTION II. DESCRIPTION OF THE APPARATUS. 

The phenomena described in the present paper are best seen 
by the method of stroboscopic observation. It is possible though 
somewhat inconveniently to observe them when the ripples are 
excited in the usual manner by a low frequency fork, which is 
electrically maintained. A dipper is attached to one of the prongs 
and touches the suifacc of the water and when the fork is set in 
vibration it excites ripples on the surface. The tank containing 
water is a rectangular vessel of => cm. depth, with a glass bottom. 
To obtain intermittent illumination two slits are attached to the 
prongs of the fork, so that they allow light to pass through them 
once for every vibration and this intermittent light is passed 
through the bottom of the tank. This arrangement, however, 
gives an insufficient amount of light. In addition to this diffi- 
culty it is not possible to control and vary the amplitude of 
motion of the dipper and its frequency in the manner desirable 
in the present investigation. These difficulties are eliminated by 
the use of Professor Fleming's motor vibrator as improved by 
Professor C V. Raman. The utility of this instrument for the 
study of vibrations cannot be overestimated. 

The apparatus can be seen in fig. I of Plate XI. To the axle 
of the motor a circular wheel is fixed and to this wheel a brass disc 
carrying a slot and a movable pin is fixed. This pin actuates an 
oscillating lever, \\hicli moves vertically up and down. To the 
end of this lever a dipper is attached, which excites the ripples on 
the surface of the liquid contained in the tank. The tank also is 
shown in the figure. It is a rectangular vessel with a glass bottom. 
To the other end of the axle of the motor a circular disc with a 
slot cut in it is attached. I/ight from an electric arc is allowed to 
pass through this hole and is reflected up by a mirror through the 
bottom of the tank and then by a second mirror projected on a 
screen. For every revolution of the motor, the light passes once, 
which makes the ripple pattern stationary. The dipper oscillating 
in the tank makes approximately a simple harmonic motion and to 
avoid any side way motion it is curved into a circulnr arc of correct 
radius. In order to be sure that the observed phenomena are not 
due to any harmonics present in the vibration of the dipper, the 
experiment is also repeated with a dipper attached to the prong of 
an electrically maintained fork, the intermittent illumation being 



178 J. C. KAMESVARA RAV. 

obtained by allowing the beam of light to pass through the 
stroboscopic disk of the motor, the speed of the latter being 
suitably regulated. 1 

It is also possible to observe the profile of the waves in the 
following manner. The intermittent flashes of light from the arc 
instead of being sent up through the bottom of the ripple tank are 
allowed to fall upon a white cardboard screen, the straight edge of 
which is viewed by the oblique reflection at the surface of the 
water contained in the tank. The edge appears in the form of a 
fixed wave which corresponds more or less closely with the actual 
wave-profile. The proper positions being chosen for the observer's 
eye and for the edge of the screen, satisfactory observations of 
the wave-profile may be made visually. The crests in the wave- 
profile correspond to the bright lines on the screen and the 
hollows to dark spaces. 

SECTION III. OBSERVATIONS OF THE FORM OK RIPPLES OF 
LARGE AMPLITUDE. 

i the amplitude is small, we see on the screen for all 
hs a scries of concentric circles at equal distances, corres- 
ponding to single crests in the waves, but when the amplitude is 
made great (say 2 mm.) the simple character of the waves is at 
once altered. For values of A up to about 1*5 cm. the waves remain 
simple circles, but if the amplitude is made very great the first 
few wave.s divide. When A lies between 1*5 and .j cm. the circles 
become double, corresponding to two crests in each wave-length. 
For still greater wave-lengths the waves become triple and 
complex. At this stage it may be pointed out, that the word 
ripples is used in the popular sense and not in the sense used by 
Kelvin. vSonie photographs of the wave patterns are given in 
figs, i toO, Plate XII. The figs. I, 2, 3 correspond to a constant 
amplitude, but of increasing wave-length and figs. 4, 5, 6 corres- 
pond to a different amplitude and increasing wave-length. It 
will be seen from the figures that with the increase of wave- 
length the doubling of the crests becomes more prominent. 



1 The amplitude of the oscillator can be adjusted by altering the position of 
the pin in the brass slot and the wave length can be altered by altering the fre- 
quency of the motor, which can be managed by including a rheostat in the circuit 
of the motor. 



J. C, K. RAV. 



PLATE X1L 





Fig 4. 



Fig. 2. 





Fig. 5, 



Flg. 3. 





Fig. 6. 



Ripples of Finite Amplitude. 



On Ripples of Finite Amplitude. 179 

Another feature that can be observed in these photographs is the 
change of form with increasing distance from the source. With 
increasing distance, the amplitude becomes smaller and consequent- 
ly the circles become dim and in the case of the double pattern, 
one of the circles gradually becomes less prominent. This can be 
seen very well in figs. 3 and 6. Also, at a great distance the 
curvature becomes smaller, and consequently the bright, narrow 
circles near the centre broaden out slightly (figs, i and 4). 

The change of the form of ripples with gradually decreasing 
depth of the liquid was also studied. Till the depth is reduced to 
a few millimeters no change is observed ; the form then gradually 
becomes simpler. The critical depth at which the change of form 
occurs is, as might be expected, less for small wave-lengths and 
greater for the larger wave-lengths. 

A curious anamolous effect is observed if the water be run out 
of the ripple tank while the dipper continues in oscillation ; the 
wave-form then becomes simple, rather suddenly, and if we stop 
running out the water at that critical moment, we get sometimes 
simple crests and sometimes double crests. This effect is ob- 
served only when the amplitude is not very large, and is apparently 
connected with changes in the form of the contact surface between 
the dipper and the liquid. Usually, however, when the running 
out of the water from the tank is stopped, the form of the ripples 
corresponding to the particular depth is at once obtained. 

We now proceed to explain the phenomena given in the 
previous section mathematically. First we take up the simpler 
case of plane waves. 

SECTION IV. MATHEMATICAL THEORY FOR PLANE WAVES OF 

FINITE AMPLITUDE. 

lyet the axis of x be in the direction of propagation of waves 
and the axis of y vertically upwards and the undisturbed level of 
the liquid as the plane y=o, and let the depth of the liquid be h. 
Then for irrotational two dimensional wave motion, the velocity 
potential satisfies the equation. 



Tl ...................... 

1 dy % 

d<t> 

7r- = o when y= h ............ (2) 

dy 



180 J. C. KAMESVARA RAV. 

This condition denotes that the velocity at the bottom is zero 
and the pressure condition at the surface gives 



dp |/<M* /d*\* 

ar* ife) +) 



taking the curvature to be small, which is true for pretty long 
waves. Since the disturbing force is simple harmonic, we can 
suppose the following expression for <j> 

S II 

<t> = 2 cosh s m (y + h) sin sm (ct x) ........ (4) 

which satisfies conditions (i) and (2). 
We have also the condition 

dp dy 

- = and for y = o 
dy m 

this gives 

i s - /; 
y = - ^ a H sinh smh cos sm (ct x) ............. (5) 

To determine the constants a l9 a fl , a^. . . . we substitute the 
values of <j> and y in the pressure equation (3) and equate the co- 
efficients of cos sm (ct - x) to zero, we get equations for the deter- 
mination of c t a^ a s .... etc., a l itself being arbitrary. For prac- 
tical purposes it is sufficient to take only three terms as the co- 
efficients 4 , # fi , etc., become negligibly small. 

Thus we can take 
< = #! cosh m (y + h) sin m (ct x) + # 2 cosh zm (y + h) sin 2m (ct x) 

+ 3 cosh 3m (y + A) sin 3^ (ct x) 
and 



= - <! #, si 



sinh mh cos fw (c/ #) + 2 sinh 2wA cos 2m (ct x) 

sinh 3wA cos 3w (c 



Substituting these values of <J> and y in the pressure equation (3) 
we get 

- + - li sinh wA cos w (ct - x) -h ( - - ^ - |2 s i a h 2mA cos 2m (c^ x) 
pc cj l \ pc c) ^ 

+ 1 2 - + j rt 8 sinh 3mA cos 3w (ct x) 
\ pc c ) 

= me [a l cosh m (y + A) cos m (ct x) + 20 2 cosh 2m (y 4- A) cos 2m (ctx) 

cosh 3m (y + A) cos sm (ct x) ] 



On Ripples of Finite Amplitude. 181 



-- [{a, cosh m (y + A) cos m (ct x) + 2,a % cosh 2m (y + A)cos 2m(ct-x) 

+ 303 cosh 3m (y + A) cos 3m (d-#)}* 

+ (d, sinh m (y + h) sin m (d-^)42a^ sinh 2m (y + h) sin 2m (ct x) 

+ 303 sinh 3m (y + A) sin 3m (ct -#)}*]. 

Expanding the right hand side of this equation in powers of y 
and substituting the value of y as given by equation (5) we get 

- * 



( + ~ i a z s * n k m h cos w (c< #) + I - + * W sinh 2mA cos 2m (ct - x ) 
pc cj \ pc cf 

1 ^ - + ja s sinh 3mA cos 3m (ct x), 
Y p c c / 



s? # 
- 
2 ! 



r 

mc| a, cosm (c*-#) i + (A-f 2 sinh smh cos sm (d- 
J^ 



w*( <-=3 a \* | 

- I A -f 2 sinh smA cos sm {ct - #) ) + ...... > 

4!V s-i c / * 

+ 2a, 2 cos 2m (ct - #) ) i + i ^-( A + 2 sinh smh cos sm (d - x) ) 
i 2! V ,,, c / 

-- i ( A ^ S sinh smA cos sm (c/ ) I -h - ...... 

4* V s=l ^ / 

-- 3a 3 cos 3m (ct-x) $ i -f^ (A->- 2 ~ sinh sm/t cos sm (ctx) \ 

C 2 ! \ sr sj C / 



-- 1 a. cos m (c/ jt) j i + ( A-f 2 s sinh smA cos sm (cl x) 1 

2L ' 2!\ s=i C J 

+ (h + 2 ~ sinh smA cos sm (c^ ^) ) -> .......... > 

4-\ s-i ^ / > 

^ 2d 2 cos 2m (cJ A;) J i +5 - LI ^ ^ 3 _f s i n h smA cos sm (ct x) I 

( 2! \ sas , C / 

+ ^ ~f A + 2 sinh smA cos sm (c/ A;) 1 

2! \ <=,, C / 

C f^m)*/ <! ~ 3 a 
f 3a s cos 3m (c/- A:) ) i + I A 4 2 ~ sinh smA cos 

1 2 ! \ SS g\ C 

+ 'j 3 - s sinh smA cos sm (c<-*)V +....( T 

s.i C / ' J 



sm (c*- 



sm (ct 



182 J. C. KAMESVARA RAV. 

-- 1 a \ i n m (rt x ) ) m (A 4 % ~ sinh smh cos 

2 L C \ ;=SI C 

"*" ~~t \ ^ "*" ^ ~~" s * n k sm ^ cos sm ( c ^ ~~ *) ) + ...... c 

420 2 sin 2m (c/-#) J 2ml h+ 2 sinh sm& cos sm (d- 

C V i = i C 

(2m) 6 / Si=3 # \ 8 

4 - pi A -- % sinh smA cos sm (ct #) I 4 

43^3 sin 3w (c/-#) < 3^! h + 2 sinh smh cos sm (C/ 

* ^ %:= I ^ 

(3m) ; V ^ a s . ' \ a ) T 

4- - { h 4 2 sinh sm cos sm (ct x) I 4 . . . . S I 

3 \ S3.I C / i J 

Equating the coefficients of cos m (ct - #), cos 2iw (c^ - x) and 
cos 3^ (ct - x) to zero, in the above equation we get the following 
approximate equations for the determination of c, a^ and a, 69 a y 
itself being arbitrary. 



4-1 sinh mA me cosh mA - a? sinh* mA cos mh 

pc cj 8 c l 

m*# 2 sinh mA sinh 2mA 4 m* a^ cosh 3 mA = o (0) 

(- -I- -\ a* sinh 2mA 2mca 2 cosh 2mA 2 a,*a sinh* mA cosh 
pc c) c 

a^a* sinh mA sinh 2mA cosh 2mA 4- a fa* sinh mA sinh 

2C 2 C 

m^ IM* m^ 

fli* sinh* mA 4 a 2 S sinh* mA = o (7) 

2 4 c l// 

(- 4 - I a sinh 3mA 3mc a s cosh 3mA - yfra\a* sinh mA sinh 2mA 
pc c/ 2 

'i*i 8 27 Ml 8 

0j*fl8 sinh* mA sinh 3mA aa sinh* mA cosh 3mA 

2C 4 

4 - ai*a s sinh mA sinh 4 mA "^ a 2 8 a 8 sinh 2mA sinh 3mA cosh 2mA 
c c 

27m 8 . , 9 , , , 15 m 3 tt . , , . , 
L a<?a b smh* 2mA cosh 3wA 4 -^ 8 *d 3 smh 5mA smh 2mA 
4 c 2 c 

2flo*tf 1 cosh mA sinh mA sinh 2mA 4 - a L a^ sinh mA sinh 3wA 

C 2 C 

+ afa* sinh 3mA sinh 2mA 4 4 a^a^ sinh 4mA sinh 3mA = o. ... (8) 

c c 



On Ripples of Finite Amplitude. 18 

From equation (6) we get 

* = ( + j[ 1 tanh mh 2 # i* sinh* mh - - m*a tanh mA sinh 2mA 
V p ml 8 c 2 



cosh 



2 - r-T 
cosh mh 

From equation (7) we get 

m * * - t-* i m * 
a.* sinh* mh -- 

4 



^ 

c 



' (^)si 

V pc c/ 

- a,* sinh mh sinh 2mA cosh mA + - a^ sinh mA sinh 3mA. 

2C 2 C 

(10) 

From equation (8) we have 

si Ml 

- w*fl,fl 2 sinh mA sinh 2mA + a } a.* sinh* 2mh 
2 c 

3m 8 . , , . , , 
~ a*a<s sinh mA sinh 3>nA 
2 c * 



(- + - 1 sinh 3mA $mc cosh 3mA a,* sinh* mA sinh 3mA 
pC C/ ' 2C 

27 m 8 Q . , , t 
- a,* sinh* mA cosh 
4 c 

+ 2 a,* sinh mA sinh 4mA - <*/ sinh 2mA sinh 3mA cosh 2mA 
c c 

- a * sinh* 2mA cosh 3mA + m 8 ao* sinh 5mA sinh 2mA. 

4 c 2 ~ 

(II) 

Hence 

y = sinh mA cos m (ct x) 

c 

(a,* sinh* mh j sinh 2mA cos 2m (ct x) 
2 4_7 

(/iTVfi* \ / 

4 gf ) sin 2mA - 2mc* cosh 2mA - m 8 aft 2 sinh* wA cosh 2mA + 
. P ' \ 

. . -f- - sinh* 2mA - sinh mA sinh 3mA ) 
4 2 / 



184 J. C. KAMESVARA RAV. 

IW 8 

- nPa x a % sinh wA sinh 2mA + a { af sinh* 2mA 
2 c 



sinh mA sinh 
2 c 



(- + g ) sinh 3mA ^mc a cosh 3mA m 8 a, 4 
P / o 

(- a. 2 sinh* mA sinh 3mA + - + I 
2 4 / 

Putting ~ sinh mA = a, the above formula becomes 

- , \ sinh 2mA cos 2m (ct-x) 

2 4 smh* mA/ 
y=a cos m(ctx) + y .. ft 

/4- W \ . - , a 1 T 

I - 1 + g l sinh 2mA 2mc* cosh 2mA 

o / , , i o i 3 cosh 3mA \ 

m*a*\ 2 cosh 2mA + cosh* mh - ^ 1 

V 2 cosh nth / 

<5 a t_ j 9 o/sinh* 2mA 3 . , ,\ 1 

{ cm*aa.} sinh 2wA - aaJ'm*\ : ; sinh 3mA I } 

12 V sinh mA 2 / ) 

x sinh 3mA cos 3m (ct x} 

I- -*-g jsinh 3mA-3mc' z cosh 3mA c a m 8 a 2 ( j sinh 3mA 

27 , . 3 sinh 4mA \ 

4- cosh 3 mA-^- - 1 1 

4 2 sinh mA / 

(2*7 
2 sinh 4mA sinh 3mA 4- ~- sinh 1 mA cosh 3mA 
4 

sinh 5wA sinh 2m/t 

2 

(12) 

For infinite depth this formula becomes 

a*c* cos 2m (ct-x) 

?. v ' 



2 
cos w (c/ ^) + 



On Ripples of Finite Amplitude. 



(4Tw* \ 
3 - 4- g I 



t i R 

2 we* + Jm 8 V 



where 



and c 

p W 

From this formula, the curves given below have been con- 
structed to show the change of form as we increase the wave 
length. 

In Fig. 7, a -2 and w = j 

y~-2 cos m (ct x) + -16 cos 2m (ct x) + -03 cos 3m (ct x) 

In Fig. 8, a = -2 and w = 2-5 

y = -2 cos m (c #) -12 cos 2m (c* - x) -ooi cos 3m (cf - %) 

In P'ig. 9, a = -2 and w = 2-25 

y = -2 cos m (cl x) -02 cos 2m (ct- x) -07 cos 3m (c< ~ %) 

In Fig. 10, ^ = -2 and m = 2. 

y = -2 cos m (ct = x) -i cos 2m (c< %) -05 cos 3m (ct x). 



FIG. 7. 



FIG. 8. 



FIG. 9. 



FIG. 10. 



The figures given in figures 7 to 10 correspond closely with 
those obtained experimentally for the same amplitudes and wave 
lengths. For practical purposes a depth of 4 or 5 cms. serves our 
purpose. The formula given above shows how these changes of 



1 86 J. C. KAMESVARA RAV. 

form come only between certain wave lengths, and how far large 
gravity waves, a. z and 3 become small and then the form becomes 
cycloidal, 1 as obtained by Stokes. The formula given above gives 
the form of small ripples as well as large gravity waves. 

The formula (12) shows also the change of form of the waves 
with depth. From the numerator of the coefficient of cos 2m 
(ct - x) we can see how the change is more prominent for small 
values of m, i.e. for large values of X. 

We shall next discuss the case of the diverging waves mathe- 
matically. 

SECTION V. MATHEMATICAL THEORY FOR DIVERGING WAVES. 

In this case we have to take cylindrical co-ordinates and take 
the axis of z vertically upwards and as before the undisturbed sur- 
face of the water as the plane, 2=0. 

The equations are modified as 

d*0 i d<t> dty 

aZ-^S^a?- (I3) 

^ u 

-z- = o when r = oo ( 14) 

or 

d<f> 

g^ = o when z** - h (15) 

The pressure equation becomes 



We can therefore assume 

f>= 2 a, J (s kr) cosh sk (z + h) sin s kct where = . (17) 

Sru. I A 

and from the condition ~- = we get for z = o 

a? CH 

2*=- 2 a s J (s &r) sinh s && cos s ^c/ (18) 

^ S= 1 

Substituting these values in the pressure equation (16), and 
taking only three terms into consideration as in the case of plane 
waves we get 



1 For large gravity waves the formula reduces to 

yssa cos m (ct x) ma* cos 2m (cf#)+ m*a8 cos 3m (ct~x). 

2 o 



On Ripples of Finite Amplitude. 187 

=* T 

3 s *a, J (s kr) sinh skh cos s&c/ I 
-' J 

a 6=3 >3 

= -- 2 6 J (s kr) sinh sAA cos skct + kc 2 sa s J' (&r) cosh $k (z + h) 

C S9 I S= I 

sin s kct 

** ( p* 3 T* 

j I 2 sa s J (A?) sinh s k (z + A) sin s kct \ 

2 " L.Srl J 

+ 1 S s0.s J ' (Ar) cosh s & (s + A) cos skct \\ 

Expanding the right hand side of this equation in series of z 
and substituting the value of z given by (18) we get 



i=^ f s^Tk* f 1 

S ] -- - J " (s kr) 4 2. J () ( S kr) [ s sinh s kh cos s kct 

Sal V. pC C J 

p- fob / S=3 ^ v % 

= kc a, J (^y) cos kct I I + -MA + 2 - s J (s kr) sinh s &A cos s kct 1 

L ^N i I C / 

^*/ ^=3^. \4 -| 

+ --( A + S J ( *?) sinh s kh cos s Ac/ 1 4- ... I 
4 J \ s=i c / J 

[/2M*/ ^^ a 
i -^^-l A - S J (s kr) sinh s AA cos s kct 
2 . \ 5 j C 

(2k)*( s=s3 a v \* 

+ ~l A -h 2 - J (s Ar) sinh s ^A cos s kct] + . . . 
4 v .s= i c I 

+ 3 kc J (3 kr) cos 3 Ac/ fi -I- ^-( A -h V - J (s Af) sinh s AA cos s Ac*Y 

L 2 ! \ s=sl C / 

(3 A)/ S=S 3 ^ V4 -i 

^ TTI * + 2 "" Jo ( ^) sinh s AA cos s AcA + . . . I 

4 l v i=i c / J 

-TI* 1 J'o(^)sinA^ J I^|-(A+ S 3 ^ s J (s *r) sinh s AA cos s AcfV 
^ L. 2 \ i=51 c y 

A*/ Sss3 a, \* \ 

+ 7}\ h + 2 ""Jo ( s ^)sinh sAAcossAcH + ... ( 

4 V s=i c y I 

r /2 k\^/ & ~1 a, 

+ 2a* J' Q (2kr)sm 2 kct] -L^\L(h^ S* - J (s Ar) sinh skh cos sAc/ 

^ 2 I \ S =BI C 

+ ( ^rr-(h + 'i 3 - J (s kr) sinh s kh cos s kctY + . . .1 
4 ! \ si ^ / > 

+ 3s J'o (3 *r) sin 3 kct \ i + ( ^(h + T - J (s kr) sinh s 6A cos s keif 

t 2 ! \ 5=sr C / 

77- ( A + S ^ J (5 Ar) sinh 5 kh cos s kct] + . . . 1 | 

I- V 4=1 C / J [ 



(3*)* 
' 4! 



188 J. C. KAMESVARA RAV. 

A* I ( t 5 ~3 d \ 

-- 1 u i Jo (* r ) s * a Ac/ f A(A+ 2 J (skr) sinh skh cos 5 ken 

2 L * \ sas| C / 

8/ s-Sfl \ I 

+ .1 h + 2 J (s kr) sinh s AA cos 5 kct) + . . . I 
3^V s=i c J * 

! SSS * 3 0s \ 

2 k (h + 2 J (5 Ay) sinh s kh cos s kct 1 
S= I C / 

S=t 3/z \* 1 

A + 2 - J ($ Ay) sinh * AA cos s JfecM --... 



- --... 

3 ^ V s=i / " 

-f 3^8 J (3 ^^) siii 3 kct \ 3 &( h + S - J (s &r) sinh s ^/r cos s kct j 

\ Saal C / 

/ s =3 /z \8 > n* 

h - 1 " 2 - Jo (^ r) sinh .s kh cos s *c* I + . . ( I 
.s=i c / ' J 



Equating the co-efficients of cos kct, cos 2 kct and cos 3 kct to 
zero, we get the following approximate equations for the deter- 
mination of c y a. 2 and a. s . 



J " (kr) sinh 6A+* J (kr) cosh M-c J (^r) cosh kh 

pC C 

k* \ A >a 

-- <** (Jo tkr)}* sinh* kh-- a, J (&r) J (2 ^r) sinh mh sinh 2 m/i 



2C 

yfe* 
<** Ji (^) Ji (2 kr) cosh wA cosh 2 wA = o .................. (IQ) 

2, 

% K 

Jo" ( 2 ^ r ) s ^ nn 2 ^A ^ - 2 Jo (^kr) sinh 



- 2 c a g J (2 Ar) cosh 2 AA - A^a^f J (Ay ))* sinh* AA 

+ ~ < 2 {J, (Ar)pcosh a AA = o .............. (20) 

4 

q'/A* /? 

- - - a s J ' (3 Ar) sinh 3 AA + ^ a 8 J (3 Ay) sinh 3 kh 

pc c 

~ 3 kc a?, J (3 Ay) cosh 3 AA - - a } a . J (Ay) J (2 Ay) sinh AA sinh 2 AA 

+ etc. . . = o ...................... (21) 

From (19) we have 



.^li^Jtanh AA-h|tanh **-* a.MJo (*r)}* sinh *A tenh ** 

P Jo 



1 cosh ^^ cosh 2 AA ^ ^ a 2 J (2 Ay) tanh AA sinh 2 mA. 



T /\ i 
J o (&*) * c 

Since J " (kr) - J, (ftf) + i J, 



On Ripples of Finite Amplitude. 189 

We have as a first approximation 

\ P Jo(^ r ) 



P p Air J (Air)* ft 
From (20) we have 

5 ftV (Jo (kr)}* sinh* kh + - ] -{ J, (r)J cosh* ftA 
a ,_ 4___ ^ 4 

_4 j^r ^ 2 f j s j an 2 kh+**~ J (2 ftr) sinh 2 ftA 

^c c 

2 ftc J () (2 ftr) cosh 2 ft 
From (21) we have 

- fl,02 J (ftr) J (2 ftr) sin ftA sinh 2 A/i 

fl,a l? J, (ftr) (J 9 ftr) cosh ftA cosh 2; 
Jo" (3 ^) sinh 3 ftA f ^ J (3 kr) sinh 3 ftA 

o 

3 Jo (3 ^ r ) cosn 3 kh - a { a^ J (ftr) J (2 ftr) sinh ftA smh 



Hence 



J sinh ^A J (^r) cos kct 
c 



Ck<i n i T 

k*af f J (*')}* sinh 2 ^ + ^L {J, (kr)}* cosh 2 A* J 

i sinh 2 &/* J (2 &r) cos 

_ ll*! J/ ( 2 jfer) sinh 2 *A -I- ? J (2 yfer) sinh 2 ^A 

/)C C 

2 &c J (2 A:r) cosh 2 
^ ^1^2 Jo (^ r ) Jo ( 2 ^ r ) s * n k kh sinh 2 A:A 



I a,a J, (^r) J, (2 &r) cosh ^A cosh 2 k/i 



-- a, 2 ( ---- ) x sinh 3 &/* J (3 /) cos 3 

2 



Putting sinh ft^ = a, and simplifying we get 



c 
z=*a J (Ar) cos 



f 3 c*ft V {Jo (kr) }* -f- - cV ( J, (ftf ) }* coth* kh] sinh 2 ftA cos 2 ftc/ 

L4 _ 4 ____ _____ __ -J. _______ 



sinh 2ftA + g sinh 2ftA-2 ftc* cosh 2 

/> J (2 ftr) 



190 J. C. KAMESVARA RAV. 

^caa% J Q (kr) J ^ (2 kr) s'mh 2 kh - ca a% J, (kr) J, (2kr) coth&Acosh 2kh 
x sin 3 kh J (3 kr) cos $kct 

_ 2Z^! J/ ( 3 kr) sinh 3 fc* + g J (3 &0 sinh 3 Kh - 3 c*& J (3 *r) cosh 3 ^A 
P 

For infinite depth this formula reduces to 



[o 
- 
^ 



J2nzs 

V p J Q (2kr 
J (Kr) J (2 /ey) -ca * 8 J, (*r) J, 



c 



cos 



The asymptotic values of J (kr) is / - cos ( kr - - V and that of 



Substituting these values in the above formula, we get 

y 
-4- COS ( ^ - - I COS fotf + J 
^ V 4/ 



- V(i --2 sin 2 kr) 



cos 2 



--etc. 

The above formula shows that the amplitude of the second 
harmonic decreases inversely as the distance from the origin, 
whereas the fundamental decreases inversely as the square root of the 
distance. Hence the wave form would tend to become simpler with 
the increase of distance from the source. This is what is also 
found experimentally. 

SECTION VI. DEPENDENCE OF THE VELOCITY OF RIPPLES ON 
AMPLITUDE AND DEPTH OF LIQUID. 



p mJ 

In the case of waves of finite amplitude, a becomes a positive 
fraction. The value of a can also be easily deduced 1 for large waves 

i Lamb's Hydrodynamics (4th edition), p. 410. 



On Ripples of Finite Amplitude. 191 

by taking only one term in the series and by using the method of 
successive approximation. 

In order to find the velocity of propagation experimentally, 
it is necessary to find the absolute frequency of the motor. This 
can be easily measured by means of an electric tachometer. But 
for purposes of comparison it is not necessary to find the absolute 
frequency. Since the velocity is proportional to wave-length , if the 
frequency remains constant, it will be sufficient to measure only 
the wave-lengths. The amplitudes of oscillation are measured by 
means of a travelling microscope or simply by means of the scale 
in the eye piece of a low power microscope. The wave lengths are 
measured on the screen and knowing the ratio of the length of a 
line on the surface of water and its image on the screen we can 
find the absolute value of the wave length of the ripples. In 
taking these measurements, it is absolutely necessary to keep the 
frequency of the motor constant, or otherwise the results are not 
comparable. This can be easily secured by means of the rheostat 
which is included in the circuit of the motor. The frequency can 
be read off directly from the electric tachometer. Some observa- 
tions made in this manner show that the velocity of long waves 
increases with the amplitude as indicated by the theory. A more 
detailed comparison between theory and experiment is at present 
being carried out. 

Neglecting for the time being the effect of amplitude we get 



p W 

This form is well known and it is sufficient here simply to 
indicate that this value of c* agrees with the experimental results 
when the amplitude is very small. From the formula it can be 
seen that for small wave lengths the velocity continues to be con- 
stant till a very low depth of a few millimeters is reached. But 
for large wave lengths, i.e. for small vales of w, the increase in the 
value of c % becomes apparent even for greater depths. This result 
is in agreement with what is actually observed. 

SECTION VII. OUTWNE OF FURTHER RESEARCH TO BE 

CARRIED OUT. 

The preceding section describes the work that has been carried 
out so far. There is much that still remains to be done. In the 



192 J. C. KAMESVARA RAV. 

present investigation regarding the form of ripples, we have not 
taken into account the effect of viscosity. The chief effect of vis- 
cosity is to diminish the amplitude and this in its turn affects the 
form of the wave. It appears also likely that the variation of the 
velocity with amplitude may have a marked effect on the group 
velocity of ripples of finite amplitude. It will be interesting to 
investigate this experimentally. In the theory of group velo- 
city usually given, the effect of finite amplitude on the speed of 
the waves is not taken into account, and it seems possible that this 
may considerably influence the results. As we have seen, the effect 
of increasing the amplitude beyond a certain point is to cause the 
waves to divide and it would be interesting to make direct mea- 
surements of the amplitude of the waves in such cases and to 
compare the same with theory. The dependence of ripple form on 
depth, especially in those cases in which the tank is of variable 
depth also appears to merit fuller investigation. The vessel in 
which the liquid is contained might, for instance, be wedge-shaped 
or hemispherical, and so on. Experiments with other liquids might 
also be carried out. 

SECTION VIII. SYNOPSIS. 

It was shown in the previous sections how the form of ripples 
of different wave lengths changes with the amplitude and fre- 
quency. To study these forms Fleming* s motor vibrator as improved 
by Prof. Raman was used to excite the ripples. This vibrator has 
many advantages over the ordinary electrically maintained tuning 
fork, viz. the amplitude of vibration and the frequency can be 
regulated to a nicety. The bottom of the tank is made of a glass 
plate and is illuminated by intermittent light, so that the ripples 
appear stationary. 

It was found that when the amplitude is small, the ripples 
were simple undulations for all wave lengths, but for finite ampli- 
tudes, with the increase of wave length, the crests of the waves 
become double, triple, and so on. With increasing distance from 
the source, these forms become simpler, as also with decreasing 
depth of the liquid in the tank. A mathematical theory explaining 
all these phenomena is given, both for plane and diverging waves. 

It is also shown that the velocity of large waves increases 
with amplitude and the mathematical results are at present being 



On Ripples of Finite Amplitude. 193 

verified. The way in which the velocity depends upon the depth is 
ilso worked out, and there it is shown that the velocity decreases 
vith the depth, though the change is not perceptible in the case 
>f smaller waves till a very smaller depth is reached. 

In conclusion, the author wishes to express his best thanks to 
Prof. 0. V. Raman who suggested the investigation for the facilities 
ifforded and for his constant encouragement and interest during 
;he progress of the research. 



XV. The Effects of a Magnetic Field on the 
General Spectrum. 



By H. P. Waran, M,A., Government Scholar of the 
University of Madras. 



(Plate XIII, Fig. 2.) 

The complex change taking place in the source when the 
radiation is emitted in a magnetic field does not seem to have had 
the attention it deserves. Though the spectrum shows a variety 
of changes under the influence of the field, yet the study of the 
spectrum has mainly been confined to the Zeeman effect on a 
particular line isolated from the rest of the spectrum. 

Some remarkable changes observed in the case of a mercury 
discharge-tube brought the phenomenon to my notice and some 
aspects of the problem with the rare gases were experimentally 
worked out and some of the very interesting results obtained 
summarised in a paper * read before the Cambridge Philosophical 
Society. 

Since then my attention has been drawn to the fact that some 
work has already been done on the subject but without coming to 
any definite conclusions. As early as 1870 Rand Capron in his 
book on Aurora mentions some experiments with a discharge-tube 
placed in a magnetic field, and speaks of some lines being brought 
out by it. 

The latest paper* on the question has been by Norton A. 
Kent and Royal M. Frye, where a bibliography of the past work 
on the subject is also given. In this paper though the authors set 
out with the idea of elucidating the real nature of the phenomenon, 

i Proceedings, Cambridge Philosophical Society, 1920. 
^ Astrophysical Journal, Vol. 37 (1913), p. 183. 



1 96 



H. P. WARAN. 



the experimental conditions they adopted seem to have led them 
to doubtful conclusions. In the main they attribute the changes 
observed to the disruptive action of the discharge when it is 
deflected by the field into the walls of the tube, the changes in the 
spectrum being due to the products of dissociation of the glass. 
Though this conclusion is fairly justifiable from their experiments, 
yet there are a few aspects of the problem which are not so easily 
explained. In fact my observations point to the contrary effect, 




that this dissociation of the glass is purely a side issue and a 
natural consequence when powerful fields and discharges are used* 
In practice by the employment of such powerful fields and discharges 
the real effect gets masked and complicated to a considerable 
degree. To examine this effect of the magnetic field on the 
radiation, the disruptive action should be reduced to a minimum 
by the employment of a moderate field and current and even then 
the tube must be of a material like quartz not liable to suffer such 
easy decomposition arising from any slight local heating. 




E 
U- 5 



O 4> 
^ O 



UJ 




E 

a 

a. 




The Effects of a Magnetic Field. 197 

In my experiments the diameter of the quartz capillary was 
about i -5 mm. and the current about 2 milliamperes, the field 
being of the order of about 5000 C.G.S. units. The arrangement 
adopted is illustrated in Fig. I, and the results obtained point to a 
definite and positive effect of the magnetic field whose nature, 
however, is not yet quite clear. 

In the case of a mercury discharge-tube at a rather low pressure 
the four lines in the red 

\\ 6234, 6152, 6123, 6072 

appear faintly. But the effect of the magnetic field on these four is 
different. The line A 6152 is enhanced very much while the others 
decrease in brilliancy. In fact at a certain stage the line \6i52 is 
practically invisible and is then brought out very brilliantly by the 
magnetic field. vSuch an anomalous behaviour in a set of lines belong- 
ing to the same element can be attributed only to a positive effect 
of the magnetic field. It may also be mentioned here that this 
line \6i52 of mercury has already appeared abnormal, in that it is 
a line brought out when helium is present in the tube. This line 
seems to have some peculiar significance in the spectrum of mercury 
and deserves a special study. Fig. 2 (a) in the Plate XIII shows 
the effect in the case of the mercury spectrum and this abnormal 
line is marked by a dot. 

A line of argument to explain this effect led to experiment 5 
with the rare gases helium, neon, etc., and their mixtures with the 
diatomic gases oxygen, hydrogen, etc. The effect of the magnetic 
field was to enhance very considerably the lines of these monatomic 
gases, even when they are present in a pure state. When mixed with 
other diatomic gases the remarkable point is that the lines of those 
monatomic gases alone are considerably enhanced, while those of the 
diatomic and others are scarcely affected. Thus in a tube of neon and 
hydrogen the principal hydrogen lines are not affected at all, while 
the neon lines are very much enhanced [Fig. 2 (b) in the Plate]. 
Similarly when a trace of helium is mixed with hydrogen or oxygen 
and the helium lines are ordinarily not visible at all, the magnetic 
field brings out the helium lines prominently, leaving the others 
scarcely affected [Fig. 2 (c) ]. These results led to the natural infer- 
ence by analogy that such enhanced lines are the radiations 
of an atom while the others may be attributed to the molecule. 



ig8 H. P. WARAN. 

The reason why the atomic radiation should be enhanced is not 
yet known. Thus in the case of mercury the enhanced lines 
^6152 and others are evidently to be classified as atomic radia- 
tions and the unaffected lines are due to the molecules. In such a 
view it is not meant that these are absolutely pure radiations. It 
may be that every line is made up of the radiations of both the 
atom and the molecule, but that the proportions are different in 
different cases, so that we may explain the highly enhanced 
radiations as those in which the atomic radiation predominates. 
The enhancement of the argon lines noted by Kent and Frye is 
also in keeping with this explanation. 

This line of argument led to experiments with sulphur and 
iodine in a vacuum tube and as is to be expected from the complex 
of molecules and atoms in all states of aggregation which we know 
to exist there, the spectrum, under the influence of the magnetic 
field, showed a variety of marked changes. These cases are undei 
more detailed study at present. Even in the case of the diatomic 
gases the effects are not simple, but considerably complicated, 
since their atomic or molecular state and the proportionality of 
their radiations can be expected to be dependent on pressure and 
temperature. Even in the case of atmospheric air at low pressure 
some marked changes are observed identical with quartz or glas-> 
tubes, so that the question of the disintegration products of the 
glass and its complications do not arise at all. Further examina- 
tion of the phenomenon in greater detail is in progress. 

Cavendish Laboratory, 

Cambridge, England ; 

May 6th, 1920. 



XVL An Improved Type of Automatic 
Mercury Pump* 



By H. P Waran, M.A., Government Scholar of the 
University of Madras. 



(Plate XIII, Fig. land i().) 

The pump under consideration is of a German design and 
worked by the mercury falling in drops down a long fall tube 
catching small pellets of air between as per original sprengel 
principle. The pump is automatic in action since the mercury 
that fell down to the bottom reservoir is transferred back again to 
the upper reservoir and made to flow again with the help of a 
filter pump worked off the main water-supply. 

A quick acting pump of the type was urgently required in 
connection with some cathodic deposition experiments and when 
the pump was constructed as per original design it was found not 
to work at all. Further experimentation with it showed the necessity 
for a good deal of improvement to be made on the original design 
before it could be made working efficiently. 

Described in brief the original design of the pump was as 
follows : 

A wide-necked flask- like vessel A with a tubulure at the bottom 
is the main mercury reservoir and it contains about 2lb. of 
mercury. On the rubber connection from the tubulure downwards 
to the bend of the air trap tube B is placed a clamp C to regulate the 
flow of mercury. The mouth of A is closed with a rubber stopper 
through which passes the central mercury return tube R and a 
drying tube of calcium chloride. From the bottom of the air trap 
1 leads up a tube which branches upwards to the receiver 
connections and downwards as the main fall tube F. This fall 
tube is about 4 to 4^ ft. in length and terminates in a lower 



200 



H. P. WARAN. 



reservoir, a big stout test tube closed by a three-hole rubber cork 
through which a side suction tube S as well as an air inlet drying 
tube D also passes. The suction tube goes up along the side and 
connects to the top of the mercury return tube R. From the top 
bulb of this return tube /?, an exhaust tube with cock X is led through 
a drying bottle P of concentrated sulphuric acid, to the suction 
pipe of a filter pump. The tube Y to the receiver goes along 




f The dotted lines indicate portions of the original design which do not appear 
in the finally modified form of the instrument.] 

a phosphorus pentoxide tube and terminates at the receiver ( a 
discharge tube) with aside connection to the vacuum gauge. 

Now when the cock W is closed, the filter pump started and 
the cock X opened, air is rapidly sucked away fom the'whole system 
until it reaches the limit of suction of the pump, i.e. about 2 
centimetres of mercury pressure. Then the clamp C is gradually 
opened and mercury run into from the trap at T and fill up the 



Automatic Mercury Pump. 201 

bottom test tube reservoir up to the mouth of S. Then the rate 
of fall of mercury is reduced with C and cock W is opened when a 
vigorous suction of air is maintained up S and all excess mercury 
coming down F in pellets is sucked up along with alternate pellets 
of air into the tap of Q wherefrom it reaches the main reservoir A 
vertically below. Thus the mercury is maintained in automatic 
circulation and the pellets of mercury going down the fall tube do 
the exhaustion. This was the original design of the pump. 

When such a pump was made and tried it was not found to 
work quite efficiently and at first the mercury could not be got 
into automatic circulation at all. Immediately experiments to set 
it going were begun after temporarily disconnecting the air trap 
and other accessories. 

Instead of a single filter pump a pair of filter pumps were 
connected in parallel, and even though the upward draught was 
much more powerful the mercury could not be got into automatic 
circulation. Further thought on the subject suggested that the 
main essential point was the necessity to establish a very strong 
air suction through the side tube to enable the mercury to be 
sucked up as rapidly as it comes down the central main fall tube. 
To secure this the vacuum created by the filter pump should be 
high and speedy and the main drawback of the original design was 
detected to be the large capacity introduced in the air suction 
circuit. The drying bottle of concentrated sulphuric acid as well as 
the bottom reservoir were too big, and once the bottom of the tube S 
got uncovered and the full atmospheric pressure of air rushed in, it 
took some time for the pump to establish the original vacuum and a 
powerful draught to take up the mercury from the lower reservoir. 
It naturally follows that the smaller this capacity, the more quickly 
the vacuum created and the more powerful the draught. So a small 
drying tube of calcium chloride is put in between the pump and 
the suction tube, while the main drying bottle of concentrated 
sulphuric acid is put in lower down to enable the air to get dried 
before being sucked into the system. The next improvement was the 
abolition of the lower reservoir altogether wich was an improvement 
in not only that it decreased the lower capacity still further, but 
eliminated the difficulty of a 3-holed rubber cork with air leaks 
and so on at this reservoir. Instead of this reservoir the lower end 
of the fall tube was curved and made to join up the suction tube 



202 H. P. WARAN. 

with a small bulb at the beginning of the suction tube with a tap sealed 
on to it. This tap leads to the drying sulphuric acid bottle and 
the suction tube of this bottle has another tap sealed on to it 
partly for regulation and partly to close up the acid from external 
moisture when the pump is left idle. This improvement prevented the 
accumulation of mercury at the bottom owing to any maladjustment 
of the rate of fall and suction, since the moment the mercury level 
rose up in the bulb to close the air inlet, the excess was sucked 
up in the powerful draught and then returned to the main reser- 
voir through the tube R. With this modification the automatic 
mercury circulation worked up quite quickly and very vigorously 
even when only one ordinary locally made glass filter pump was 
used. The air getting mixed up with the mercury is quite dry 
and no fouling of the vacuum from vapour pressure is to be expected. 
It is very useful to observe the general precaution that the bore of 
the suction tube 5 as well as that of the fall tube should be about 
a millimetre and not more, to prevent the air escaping by the side 
of the mercury pellets. These tubes are conveniently made of 
thick walled capillary glass tubing of internal diameter about a 
millimetre. 

The vacuum air trap is as usual and is an essential adjunct 
when the highest vacuum is to be used. With these modifications 
introduced the pump works quite easily and rapidly, perfectly 
automatic in action, once the rate of fall of mercury is adjusted 
and the two taps at the bottom opened after preliminary vacuum 
of about 2 centimetres of mercury has been directly obtained by 
the filter pump. Quite a cathode ray vacuum is produced in about 
30 or 40 minutes everything going well. But working the pump 
longer, the vacuum was found not to improve further, and it was 
difficult to push the discharge tube to quite a hard condition. An 
examination in detail of what is happening in the fall tube at this 
stage shows the reason why. The fall tube must be scrupulously 
clean and when the metallic clicks of the falling mercury pellets 
are noticed it will be found that a good length of air in the fall tube 
is compressed by the falling column of mercury to quite a tiny 
disc of air of imperceptible size. This air somehow does not seem 
to escape out of the tube along with the mercury, but somehow 
escapes back into the high vacuum system, probably escaping 
gradually by the side of the falling pellets. This difficulty can be 



Automatic Mercury Pump. 203 

got over if we can prevent the air being compressed into this thin 
disc-like form. 

A suggestion to get over this difficulty is to do this latter high 
exhaustion in two stages so that in the first stage the air is not 
compressed but simply pushed into another highly exhausted space 
to produce the extreme vacuum. Following this idea the fall was 
made in two stages, the first of about 30 centimetres and the other 
of about 90 centimetres, and the tip (lower) of the first tube was 
just touching the mercury level on the reservoir on top of the 
lower tube as illustrated. In such a system, at the initial stages 
when the pressure is high, the first fall is not very effective and 
the real exhaustion takes place by the lower fall tube as usual. 
But later as the vacuum gets very high and the mercury begins to 
hammer in the lower fall tube, the rate of flow of mercury is slight- 
ly increased to secure two or three pellets going down the first tube at 
the same instant. This succession of pellets down the first fall are 
very effective and they push the remaining air from the receiver 
into the upper reservoir from which they can never escape into the 
high vacuum so long as the downward stream of mercury continues 
and is maintained permanently by the filter pump. The little air 
that collects in the upper reservoir is of course removed whenever 
it is an appreciable quantity by the lower fall tube. This is the 
most important of all the improvements and its effect is quite 
remarkable. With its help the pump exhausts to limits which it 
could not previously exhaust and in the case of one small locally 
made X-ray tube with electrodes about 2 inches apart, the discharge 
could be made to prefer an alternate air gap of about 4 or 5 inches 
with a six-inch coil. 

The above piece of work was done at the Presidency College, 
Madras, and Fig. I in Plate XIII shows the pump during its experi- 
mental stages when just got working automatically. Fig. i(a) 
is a photograph of the double stage exhaustion attachment for high 
vacuum work. 

Cavendish Laboratory, 
Cambridge, England. 



Index* 

PAGE 

Banerjee, Bhabonath Mechanical Illustrations of the Theory 

of Large Oscillations and Combinational Tones . . 37 

Braunites, Indian, the Magneto-Crystalline Properties of 87 

Christiansen's Experiment and Wave Propagation in Optically 

Heterogeneous Media .. .. .. 121 

Chuckerbutti, B. N. On the Production of Musical Sounds 

from Heated Metals . . . . . . . . 143 

Combinational Tones and Large Oscillations, Mechanical Illus- 
trations of the Theory of . . . . . . 37 

Convection, Free and Forced, from Heated Cylinders in Air . . 95 

Flute, the Theory of the .. .. .. .. 113 

Forced Oscillations of Stretched Strings under Damping pro- 
portional to the Square of the Velocity . . . . 67 

Ghosh, P. N. Some Phenomena of Laminar Diffraction ob- 
served with Mica . . . . . . . . 51 

Ghosh, R. N. On the Forced Oscillations of Stretched Strings 

under Damping proportional to the Square of the Velocity. 67 

Some New Illustrations of Optical Theory by Ripple 

Motion . . . . . . . . . . r55 

Heliometer, On a New Geometrical Theory of the Diffraction 

Fringes observed in the . . . . . . i 

Heterogeneous Media, On Wave Propagation in Optically and 

the Phenomena observed in Christiansen's Experiment . . 121 

Impact, Experiments on . . . . . . . . 109 

on Elastic Plates, the Theory of . . . . . . 165 

Kamesvara Rav, J. C. On Ripples of Finite Amplitude . . 175 

Laminar Diffraction, some Phenomena observed with Mica . . 51 

Large Oscillations and Combinational Tones, Mechanical 

Illustrations of the Theory of . . . . 37 

Magneto-Crystalline Properties of Indian Braunites . . 87 

Mechanically -Played Violins, Experiments with . , . . 19 

Mercury Pump> an Automatic . . . . . , 199 

Mica, some Phenomena of Laminar Diffraction observed with 51 
Mitra, S. K. On a New Geometrical Theory of the Diffraction 

Figures observed in the . . . . . . i 

Musical Sounds from Heated Metals, On the Production of , 143 

Pump, An Automatic Mercury . . . . . . 199 



206 INDEX. 

PAGE 

Raman, Prof. C. V. Experiments with the Mechanically- 
Played Violins . . . . . . . . 19 

Ray, Bidhubhushan Free and Forced Convection from 

Heated Cylinders in Air . . . . . . 95 

Ripple Motion, Some New Illustrations of Optical Theory by . . 155 

Ripples of Finite Amplitude . . . . . . 175 

Seshagiri Rao, K. The Magneto-Crystalline Properties of 

Indian Braunites . . . . . . . . 87 

Theory of Impact on Elastic Plates . . . . 165 

Sethi, N. K. On Wave Propagation in Optically Heterogene- 
ous Media and the Phenomena Observed in Christian- 
sen's Experiment .. .. .. .. 121 

Sounds, Musical, from Heated Metals, On the Production of. . 143 

Spectrum. General, On the Effect of a Magne tic Field on . . 195 

Strings, On the Forced Oscillations of Stretched , under 

Damping proportional to the Square of the Velocity . . 67 

Venkatasubbaraman, A. Experiments on Impact . . 109 
Violins, Mechanically Played, Experiments with . . . . 19 
Walker, Dr. G. T. The Theory of the Flute . . . . 113 
Waran, H. P. An Automatic Mercury Pump . . . . 199 
Effect of a Magnetic Field on the General Spectrum . . 195